Strong laws of large number for intermediately trimmed Birkhoff sums of observables with infinite mean

TL;DR

This paper extends strong laws of large numbers to intermediately trimmed Birkhoff sums in certain dynamical systems with infinite mean, matching rates known for independent variables, and applies to systems like interval maps.

Contribution

It generalizes strong laws of large numbers for intermediately trimmed sums to dependent dynamical systems with infinite mean, including specific conditions and examples.

Findings

Validates the laws for systems with spectral gap property

Establishes trimming rates comparable to i.i.d. cases

Applies results to piecewise expanding interval maps

Abstract

We consider dynamical systems on a finite measure space fulfilling a spectral gap property and Birkhoff sums of a non-negative, non-integrable observable. For such systems we generalize strong laws of large numbers for intermediately trimmed sums only known for independent random variables. The results split up in trimming statements for general distribution functions and for regularly varying tail distributions. In both cases the trimming rate can be chosen in the same or almost the same way as in the i.i.d. case. As an example we show that piecewise expanding interval maps fulfill the necessary conditions for our limit laws. As a side result we obtain strong laws of large numbers for truncated Birkhoff sums.

Equations

$$\underset{n\to \infty }{\lim }\frac{{\mathsf{S}}_n\chi }{n}=\int \chi \mathrm{d}\mu ,$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{{\mathsf{S}}_n\chi }{d_n}=+\infty \text{ or }\underset{n\to \infty }{\mathrm{liminf}}\frac{{\mathsf{S}}_n\chi }{d_n}=0.$$

$${\mathsf{S}}_n^{b_n}\chi$$

$$\underset{n\to \infty }{\mathrm{limsup}}\pm \frac{S_n^{b_n}-n\cdot {\int }_0^{1-b_n/n}{F}^{\leftarrow }\left(s\right)\mathrm{d}s}{\gamma \left(n,b_n\right)}$$

$$x:=\frac{1}{a_1\left(x\right)+\frac{1}{a_2\left(x\right)+\ddots }}.$$

$$\underset{n\to \infty }{\lim }\frac{{\mathsf{S}}_n^1\chi }{n\log n}=\frac{1}{\log 2}.$$

$$\int \hat{T}f\cdot g\mathrm{d}\mu =\int f\cdot g\circ T\mathrm{d}\mu ,$$

$$\Vert f\Vert \ge {\left|f\right|}_{\infty }.$$

$$\Vert \hat{T}f\Vert \le K_0\cdot \Vert f\Vert .$$

$$\text{prescript}\ell \chi :=\chi \cdot \text{mathbbm}{1}_{\left\{\chi \le \ell \right\}}.$$

$$\Vert \text{prescript}\ell \chi \Vert \le K_1\cdot \ell .$$

$$\Vert \text{mathbbm}{1}_{\left\{\chi >\ell \right\}}\Vert \le K_2.$$

$$a_n:=n\cdot \left(1-F\left(f_n\right)\right),d_n:=n{\int }_0^{f_n}x\mathrm{d}F\left(x\right),n\in \mathbb{N},$$

$$\underset{n\to \infty }{\lim }{f}_n/d_n\cdot \max \left\{a_n^{1/2+ϵ}{\left(\log \log n\right)}^{1/2-ϵ},\log n\right\}$$

$$b_n:=\lceil a_n+W\cdot \max \left\{a_n^{1/2+ϵ}\cdot {\left(\log \log n\right)}^{1/2-ϵ},\log \log n\right\}\rceil ,$$

$$\underset{n\to \infty }{\lim }\frac{S_n^{b_n}}{d_n}=1\text{ a.s.}$$

$$\underset{n\to \infty }{\lim }\frac{{\mathsf{S}}_n^{b_n}\chi }{d_n}=1\text{ a.s.}$$

$$\Psi :=\left\{u:\mathbb{N}\to {\mathbb{R}}^{+}:\sum _{n=1}^{\infty }\frac{1}{u\left(n\right)}<\infty \right\}.$$

$$\underset{n\to \infty }{\lim }\frac{b_n}{\log \psi \left(\lfloor \log n\rfloor \right)}=\infty$$

$$\underset{n\to \infty }{\lim }\frac{{\mathsf{S}}_n^{b_n}\chi }{d_n}=1\text{ a.s.}$$

$$d_n\sim \frac{\alpha }{1-\alpha }\cdot n^{1/\alpha }\cdot b_n^{1-1/\alpha }\cdot {\left(L^{1/\alpha }\right)}^{\#}\left({\left(\frac{n}{b_n}\right)}^{1/\alpha }\right).$$

$$d_n\sim \frac{\alpha }{1-\alpha }\cdot n^{1/\alpha }\cdot b_n^{1-1/\alpha }.$$

$$\psi (n)=\{\begin{array}{ll}{\left({\log }_{2}n\right)}^2& \text{ if }n=2^k\text{, }k\in \mathbb{N}\\ n^2& \text{ else,}\end{array}$$

$$\mathsf{var}\left(\phi \right):=\sup \left\{\sum _{i=1}^n\left|\phi \left(x_i\right)-\phi \left(x_{i-1}\right)\right|:n\ge 1,x_i\in \left[0,1\right],x_0<x_1<\dots <x_n\right\}$$

$$\mathsf{V}\left(\phi \right):=\inf \left\{\mathsf{var}\left({\phi }^{\prime }\right):{\phi }^{\prime }\text{ is a version of }\phi \right\}.$$

$$\mathsf{V}\left(\text{prescript}\ell \chi \right)\le {\tilde{K}}_1\cdot \ell$$

$$\mathsf{V}\left(\text{mathbbm}{1}_{\left\{\chi >\ell \right\}}\right)\le {\tilde{K}}_2.$$

$$\text{prescript}\ell \chi :=\text{mathbbm}{1}_{\left\{\chi \le \ell \right\}}\cdot \chi ,$$

$${\mathsf{T}}_n^{f_n}\chi$$

$$\underset{n\to \infty }{\lim }\frac{{\mathsf{T}}_n^{f_n}\chi }{\int {\mathsf{T}}_n^{f_n}\chi \mathrm{d}\mu }=1\text{ a.s.}$$

Full text

1. Introduction and statement of main results

We consider an ergodic dynamical system $\left(\Omega,\mathcal{A},T,\mu\right)$ with $\mu$ a probability measure and a stochastic processes given by the Birkhoff sums $\mathsf{S}_{n}\chi\coloneqq\sum_{k=1}^{n}\chi\circ T^{k-1}$ with $\mathsf{S}_{0}\chi=0$ for some measurable function $\chi:\Omega\to\mathbb{R}_{\geq 0}$. Regarding strong laws of large numbers there is a crucial difference between $\int\chi\mathrm{d}\mu$ being finite or not. In the finite case we obtain by Birkhoff’s ergodic theorem that $\mu$-almost surely (a.s.)

[TABLE]

i.e.  the strong law of large numbers is fulfilled, whereas in the case of an observable with infinite expectation, Aaronson showed in [Aar77] that for all positive sequences $\left(d_{n}\right)_{n\in\mathbb{N}}$ we have $\mu$-a.s.

[TABLE]

However, there might be a strong law of large numbers after deleting a number of the largest summands from the partial $n$-sums. More precisely, for each $n\in\mathbb{N}$ we chose a permutation $\sigma\in\mathcal{S}_{n}$ of $\left\{0,\ldots,n-1\right\}$ such that $\chi\circ T^{\sigma\left(0\right)}\geq\chi\circ T^{\sigma\left(1\right)}\geq\ldots\geq\chi\circ T^{\sigma\left(n-1\right)}$ and for given $\left(b_{n}\right)\in\mathbb{N}_{0}^{\mathbb{N}}$ we define

[TABLE]

If $b_{n}=r\in\mathbb{N}$ is fixed for all $n\in\mathbb{N}$ then $\left(\mathsf{S}_{n}^{r}\chi\right)$ is called a lightly trimmed sum process. If we allow the sequence $\left(b_{n}\right)\in\mathbb{N}_{0}^{\mathbb{N}}$ to diverge to infinity such that $b_{n}=o\left(n\right)$, i.e. $\lim_{n\rightarrow\infty}b_{n}/n=0$, then $\left(\mathsf{S}_{n}^{r}\chi\right)$ is called an intermediately (also moderately) trimmed sum process. If there exist $r\in\mathbb{N}$ and a sequence of constants $(d_{n})$ such that $\lim_{n\to\infty}\mathsf{S}_{n}^{r}\chi/d_{n}=1$ a.s. we refer to it as a lightly trimmed strong law and similarly if for an intermediately trimmed sum $\mathsf{S}_{n}^{b_{n}}\chi$ there exists $(d_{n})$ such that $\lim_{n\to\infty}\mathsf{S}_{n}^{b_{n}}\chi/d_{n}=1$ a.s. we refer to it as an intermediately trimmed strong law.

Trimming results for independent, identically distributed (i.i.d.) random variables are well studied. Mori developed in [Mor76] and [Mor77] general conditions for a lightly trimmed strong law to hold. These results have been generalized by Kesten and Maller, see [Mal84], [KM92], and [KM95].

It becomes clear from a result by Kesten, see [Kes93], that light trimming is not always sufficient. His result implies in particular that a weak law of large numbers for a lightly trimmed sum of i.i.d. random variables holds if and only if it also holds for the untrimmed sum. This rules out the possibility for a lightly trimmed strong law if a weak law of large numbers does not hold.

A special case in this context is the case of regularly varying tail variables with index between $-1$ and [math]. To state more precisely the situation we require that the distribution function $F$ (i.e. in the dynamical systems setting we have $F\left(x\right)=\mu\left(\chi\leq x\right)$) fulfills $1-F\left(x\right)\sim x^{-\alpha}L\left(x\right)$ with $0<\alpha<1$ and $L$ a slowly varying function. Here, $u\left(x\right)\sim w\left(x\right)$ means that $u$ is asymptotic to $w$ at infinity, that is $\lim_{x\rightarrow\infty}u\left(x\right)/w\left(x\right)=1$ and $L$ being slowly varying means that for every $c>0$ we have $L\left(cx\right)\sim L\left(x\right)$.

It can be easily deducted from [Fel71, VII.7 Theorem 2 and VIII.9 Theorem 1] that in the just mentioned case for i.i.d. random variables no weak law of large numbers and thus no lightly trimmed strong law holds. Furthermore, the case of i.i.d. random variables with regularly varying tails is treated by Haeusler and Mason in [HM87] and Haeusler in [Hae93], in which a law of an iterated logarithm is established. As we are here considering sums of non-negative random variables instead of Birkhoff sums, we will denote in these cases the sum trimmed by the $r$ maximal terms by $S_{n}^{r}$, i.e. we distinguish the trimmed Birkhoff sum $\mathsf{S}_{n}\chi$ from a trimmed sum of i.i.d. random variables $S_{n}^{r}$. With this notation and setting $F^{\leftarrow}:\left[0,1\right]\to\mathbb{R}_{\geq 0}$ denoting the generalized inverse function of $F$, i.e. $F^{\leftarrow}\left(y\right)\coloneqq\inf\left\{x\in\mathbb{R}\colon F\left(x\right)\geq y\right\}$, then a combination of the results by Haeusler and Mason imply that there exists a non-stochastic $\gamma$ depending on $n$ and the trimming function $b_{n}$ such that

[TABLE]

almost surely equals $1$ if $\lim_{n\to\infty}b_{n}/\log\log n=\infty$, see [HM87], and, almost surely equals a constant $M$ if $b_{n}\sim c\cdot\log\log n$, see [Hae93]. By comparing the asymptotic behavior of the norming and centering sequences $\gamma\left(n,b_{n}\right)$ and $n\cdot\int_{0}^{1-b_{n}/n}F^{\leftarrow}\left(s\right)\mathrm{d}s$ referring to [Hae93, Section 4] one can conclude that $\lim_{n\to\infty}S_{n}^{b_{n}}/(n\cdot\int_{0}^{1-b_{n}/n}F^{\leftarrow}\left(s\right)\mathrm{d}s)=1$ almost surely if and only if $\lim_{n\to\infty}b_{n}/\log\log n=\infty$, i.e. an intermediately trimmed strong law follows as a special case.

The results for the lightly trimmed case have been generalized to dependent random variables from different contexts. One of the first investigated examples of this situation is the unique continued fraction expansion of an irrational $x\in\left[0,1\right]$ given by

[TABLE]

In this case we consider the space $\Omega\coloneqq\left[0,1\right]\backslash\mathbb{Q}$, the Gauss measure $\mu$ given by $d\mu(x)\coloneqq 1/\left(\log 2\left(1+x\right)\right)d\lambda\left(x\right)$ with $\lambda$ denoting the Lebesgue measure restricted to $\left[0,1\right]$, and the Gauss map $T$ defined as $Tx\coloneqq\left\{1/x\right\}\coloneqq 1/x-\left\lfloor 1/x\right\rfloor$. The observable $\chi\colon\Omega\to\mathbb{N}$ with $\chi(x)\coloneqq\left\lfloor 1/x\right\rfloor$ and $\left\lfloor x\right\rfloor\coloneqq\max\left\{n\in\mathbb{N}\colon n\leq x\right\}$ gives then rise to the stationary (dependent, but $\bm{\psi}$-mixing) process $\chi\circ T^{n-1}=a_{n}$, $n\in\mathbb{N}$, of the $n$-th continued fraction digit. Even though a strong law of large numbers can not hold for $\mathsf{S}_{n}\chi$, Diamond and Vaaler showed in [DV86] that under light trimming with $r=1$ we have a.s.

[TABLE]

These results were generalized in [NN02] and [NN03] to other continued fraction expansions.

In [Hay14] Haynes gave a quantitative strong law of large numbers under trimming for a certain class of observables $\chi$. He considered the sum $\widetilde{\mathsf{S}}_{n}$ with $\widetilde{\mathsf{S}}_{n}\chi(x)\coloneqq\mathsf{S}_{n}\chi(x)-\delta\left(n,x\right)\max_{0\leq i\leq n-1}\chi\circ T^{i}(x)$ with $\delta\left(n,x\right)\in\left\{0,1\right\}$, i.e. the sum is trimmed by the maximal element, but depending on $x$. Then he gave an error term for $\widetilde{\mathsf{S}}_{n}\chi-d_{n}$ with $(d_{n})$ the norming sequence as above.

One way to study limit theorems in the dynamical systems setting is to prove limit theorems for mixing random variables as dynamical system are often $\bm{\psi}$- or at least $\bm{\phi}$- or $\bm{\phi}_{\textrm{rev}}$-mixing. For a precise definitions of different kinds of mixing see [Bra05]. One approach in this direction is a result by Aaronson and Nakada extending the results by Mori to $\bm{\psi}$-mixing random variables, see in [AN03], i.e. they gave sufficient conditions for a lightly trimmed strong law to hold.

In this paper we will study intermediately trimmed strong laws for dynamical systems fulfilling a spectral gap property with the exact assumptions on the system given as Property $\mathfrak{D}$ in Definition 1.2. The random variables of our main example given in Section 1.4 are at least exponentially $\bm{\phi}_{\textrm{rev}}$-mixing, which was proven in [AN05]. One approach to prove an intermediately trimmed strong law would thus be to prove a limit theorem for $\bm{\phi}_{\textrm{rev}}$-mixing systems. However, it is difficult to prove these trimming results by only assuming a certain speed of mixing. These problems are elucidated in Section 2.4.

Our method is based on proving an exponential inequality for the dynamical systems using a spectral method similar to the Nagaev-Guivarc’h spectral method for the central limit theorem, giving stronger results than exponential inequalities for mixing random variables. A sketch of the proof will be given in Section 2.1 and some of the difficulties will be discussed in Section 2.4.

The spectral gap property for dynamical systems, guaranteed by our later stated Property $\mathfrak{C}$, see Definition 1.1, is a typical assumption under which limit theorems for dynamical systems can be proven. The first statements proven in this setting were central limit theorems for Markov chains, see [Nag57], and generalizations for other dynamical systems, see [RE82] and [GH88]. Also other limit theorems have been proven in this setting as for example local central limit theorems, see [HH01], [GH88], and [HP10] for higher dimensions, Berry-Esseen theorems, see [Gou05], [HP10], and almost sure invariance principles, see [MN09] and [Gou10a].

There has also been some recent interest in limit theorems for dynamical systems with heavy tail distributions. This includes in particular convergence to a stable law. Aaronson and Denker proved in [AD01] some necessary conditions for a stable limit laws for dynamical systems and observables with heavy tails. Gouëzel proved in [Gou10b] the necessity of those conditions using the work of Sarig, see [Sar06], which is restricted to a particular class of distribution functions. These results were also generalized for dynamical systems on intermittent maps, see [MZ15]. Furthermore, Gouëzel also proved a stable limit law for observables of the doubling map, see [Gou08].

Tyran-Kaminska studied in [Tyr10] the functional convergence of normalized Birkhoff sums with heavy tailed observables to $\alpha$-stable processes.

Aaronson and Zweimüller proved in [AZ14] some stable laws and additionally a one-sided law of the iterated logarithm for mixing dynamical systems and observables with heavy tails. All these results use intrinsically transfer operator techniques. For further references concerning transfer operator methods we refer the reader to the review papers [Gou15] and [FJ03].

In contrast to the before mentioned stable laws, Carney and Nicol investigated growth rates of Birkhoff sums of non-integrable observables on a finite measure space giving some almost sure results, see [CN17]. Closely related to the behavior of a non-integrable observables on a finite measure space is the behavior of observables on an infinite measure space.

The pointwise convergence behaviour of Birkhoff sums with respect to iterates of the transfer operator acting on integrable observables with a finite number of poles have been studied in the context of both finite and infinite measure-preserving dynamical systems in [KKS16]. In [LM18] Lenci and Munday further investigate the question under which conditions a Birkhoff ergodic theorem holds for infinite measure-preserving dynamical systems.

In the following section we will give the precise setting and even though not every $\bm{\phi}_{\textrm{rev}}$-mixing system can be described by a dynamical system with spectral gap, our setting for dynamical systems is rather general, as many dynamical systems fulfill the later stated Property $\mathfrak{C}$, for example subshifts of finite type, see for example [PP90] or [Bal00], piecewise expanding interval maps, see [HK82] and a generalization for infinitely many partitions, see [Ryc83], and Anosov or Axiom A systems, see [BT07] and [GL08].

To prove an intermediately trimmed strong law we need additional assumptions on the observable $\chi$ given in Property $\mathfrak{D}$. As a concrete example we state later in Section 1.4 that under rather mild assumptions a system of piecewise expanding interval maps fulfills the additional assumptions of Property $\mathfrak{D}$.

As we show, our setting is rather general in the sense that there are dynamical systems for which a lightly trimmed strong law does not hold despite the fact that it would hold for i.i.d. random variables with the same distribution function. Keeping the system but using another observable with a different distribution function allows intermediately trimmed strong laws for the same trimming sequence as in the i.i.d. case, see Remark 1.13.

In Sections 1.2 and 1.3 we will present intermediately trimmed strong laws for the above mentioned dynamical systems, for general distribution functions in Section 1.2 and for regularly varying tail distributions in Section 1.3. The results for general distribution functions are almost as strong as the i.i.d. trimming results in [KS17] and for the regular variation case we show that an intermediately trimmed strong law holds for the same trimming sequence $\left(b_{n}\right)$ which in the i.i.d. case can be derived from [HM87].

As side results we obtain limit theorems for sums of truncated random variables, see Section 2.2. Namely, for $\chi:\Omega\to\mathbb{R}_{\geq 0}$ and a real valued sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$ we consider the truncated sum $\mathsf{T}_{n}^{f_{n}}\chi\coloneqq\sum_{k=1}^{n}\left(\chi\cdot\mathbbm{1}_{\{\chi\leq f_{n}\}}\right)\circ T^{k-1}$. Limit results for these truncated sums have been of recent interest for i.i.d. random variables, see [Nak15] and [GK11].

In Section 2 we will give the main ideas of the proof, explain the differences to the i.i.d. case, and give the structure of the rest of the paper.

1.1. Basic setting

In the following we will define our two main properties. The first, Property $\mathfrak{C}$, restricts to dynamical systems with a spectral gap property and is part of our next property, Property $\mathfrak{D}$. Property $\mathfrak{D}$ is a property on the dynamical system and additionally on the admissible observables allowing us to state and prove intermediately trimmed strong laws.

Definition 1.1** (Property $\mathfrak{C}$).**

Let $\left(\Omega,\mathcal{A},T,\mu\right)$ be a dynamical system with $T$ a non-singular transformation and $\widehat{T}:\mathcal{L}^{1}\to\mathcal{L}^{1}$ be the transfer operator of $T$, i.e. the uniquely defined operator such that for all $f\in\mathcal{L}^{1}$ and $g\in\mathcal{L}^{\infty}$ we have

[TABLE]

see e.g. [KMS16, Section 2.3] for further details. Furthermore, let $\mathcal{F}$ be a subset of the measurable functions forming a Banach algebra with respect to the norm $\left\|\cdot\right\|$. We say that $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ has Property $\mathfrak{C}$ if the following conditions hold:

• •

$\mu$ is a $T$-invariant, mixing probability measure.

• •

$\mathcal{F}$ contains the constant functions and for all $f\in\mathcal{F}$ we have

[TABLE]

• •

$\widehat{T}$ is a bounded linear operator with respect to $\left\|\cdot\right\|$, i.e. there exists a constant $K_{0}>0$ such that for all $f\in\mathcal{F}$ we have

[TABLE]

• •

$\widehat{T}$ has a spectral gap on $\mathcal{F}$ with respect to $\left\|\cdot\right\|$, see Definition 3.1.

The above mentioned property is a widely used setting for dynamical systems. In particular it implies that the transfer operator has $1$ as a unique and simple eigenvalue on the unit circle which implies an exponential decay of correlation. We will give proofs of these properties in Section 3.1.

However, in order to state our main theorems we need additional assumptions on the observable $\chi$ acting on a system fulfilling Property $\mathfrak{C}$.

Definition 1.2** (Property $\mathfrak{D}$).**

For a Banach algebra $\mathcal{F}$ and for a fixed measurable function $\chi:\Omega\to\mathbb{R}_{\geq 0}$ we set, for all $\ell\in\mathbb{R}_{\geq 0}$,

[TABLE]

We say that $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ has Property $\mathfrak{D}$ if the following conditions hold:

• •

$\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfills Property $\mathfrak{C}$.

• •

There exists $K_{1}>0$ such that for all $\ell>0$,

[TABLE]

• •

There exists $K_{2}>0$ such that for all $\ell>0$,

[TABLE]

Even though a lot of dynamical systems fulfill Property $\mathfrak{C}$ as mentioned in the introduction, it is not immediately clear for which observables $\chi$ they additionally fulfill Property $\mathfrak{D}$. However, it turns out that for piecewise expanding interval maps the assumptions on $\chi$ are rather weak, see Section 1.4.

1.2. Results for general distribution functions

The following theorem provides us with a method to find a trimming sequence $\left(b_{n}\right)$ if the distribution function $F$ is given. Before stating this theorem we define $\left\lceil x\right\rceil\coloneqq\min\left\{n\in\mathbb{N}\colon n\geq x\right\}$.

Theorem 1.3**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$ and let $\left(f_{n}\right)_{n\in\mathbb{N}}$ be a sequence of positive real numbers tending to infinity. Fix $0<\epsilon<1/4$ such that for

[TABLE]

we have

[TABLE]

Then there exists $W>0$ independent of $\epsilon$ and $(f_{n})$ such that for

[TABLE]

$n\in\mathbb{N}$, we have

[TABLE]

Remark 1.4*.*

This theorem is the equivalent to [KS17, Theorem B] for the setting of dynamical systems. In [KS17] we also give an example how to find a proper trimming function for a given distribution function $F$.

As a corollary we obtain that an intermediately trimmed strong law under these conditions always holds.

Corollary 1.5**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$. Then there exist a sequence of natural numbers $\left(b_{n}\right)_{n\in\mathbb{N}}$ with $b_{n}=o\left(n\right)$ and a sequence of positive reals $\left(d_{n}\right)_{n\in\mathbb{N}}$ such that

[TABLE]

Remark 1.6*.*

We would like to point out that the sequence $\left(d_{n}\right)_{n\in\mathbb{N}}$ is not necessarily asymptotic to the sequence of expectations $\left(\int\mathsf{S}_{n}^{b_{n}}\chi\mathrm{d}\mu\right)_{n\in\mathbb{N}}$. This has been shown in [KS17, Remark 2] for a sequence of i.i.d.  summands.

1.3. Results for regularly varying tails

For stating our main theorem we set

[TABLE]

Then our main result reads as follows:

Theorem 1.7**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$ and let $\chi:\Omega\rightarrow\mathbb{R}_{\geq 0}$ be such that $\mu\left(\chi>x\right)=L\left(x\right)/x^{\alpha}$ with $L$ a slowly varying function and $0<\alpha<1$. Further, let $\left(b_{n}\right)_{n\in\mathbb{N}}$ be a sequence of natural numbers tending to infinity with $b_{n}=o\left(n\right)$. If there exists $\psi\in\Psi$ such that

[TABLE]

then there exists a positive valued sequence $\left(d_{n}\right)_{n\in\mathbb{N}}$ such that

[TABLE]

and $\left(d_{n}\right)$ fulfills

[TABLE]

Remark 1.8*.*

If $L\left(n\right)=1$, then the norming sequence simplifies to

[TABLE]

Remark 1.9*.*

The condition on the trimming sequence (7) seems very technical at the beginning. If we choose $\psi(n)=n^{2}$, then $\psi\in\Psi$ and $\log\psi\left(\left\lfloor\log n\right\rfloor\right)=2\log\left\lfloor\log n\right\rfloor$ and thus (7) can also be written as $\lim_{n\to\infty}b_{n}/\log\log n=\infty$.

However, this more general condition allows us to also consider $\left(b_{n}\right)$ which are more complicated and particularly not monotonic. For example one can set

[TABLE]

and consider the sequence $(b_{n})$ fulfilling $\lim_{k\to\infty}b_{2^{2^{k}}}/\log k=\infty$ for $k\in\mathbb{N}$ and, additionally, $\lim^{\star}_{n\to\infty}b_{n}/\log\log n=\infty$, where the last limit $\lim^{\star}$ reaches over all $n\in\mathbb{N}$ not taking the values $n=2^{2^{k}}$ with $k\in\mathbb{N}$. Then $(b_{n})$ fulfills condition (7) as well.

For the case of i.i.d. summands it follows from [Hae93] that the convergence in (8) also implies $\lim_{n\to\infty}b_{n}/\log\log n=\infty$. This implies that our bound is nearly optimal.

1.4. Example: Piecewise expanding interval maps

For our main example, we first define the space of functions of bounded variation. For simplicity, we restrict ourself to the interval $\left[0,1\right]$ and let $\mathcal{B}$ denote the Borel sets of $\left[0,1\right]$.

Definition 1.10**.**

Let $\varphi:\left[0,1\right]\to\mathbb{R}$. The variation $\mathsf{var}\left(\varphi\right)$ of $\varphi$ is given by

[TABLE]

and we define

[TABLE]

By $BV$ we denote the Banach space of functions of bounded variation, i.e. of functions $\varphi$ fulfilling $\mathsf{V}\left(\varphi\right)<\infty$. It is equipped with the norm $\left\|\varphi\right\|_{BV}\coloneqq\left|\varphi\right|_{\infty}+\mathsf{V}\left(\varphi\right)$.

For further properties of functions of bounded variation see e.g. [BG97, Chapter 2].

Proposition 1.11**.**

Let $\Omega^{\prime}\subset\left[0,1\right]$ be a dense and open set such that $\mu\left(\Omega^{\prime}\right)=1$ and let $\mathcal{I}\coloneqq\left(I_{n}\right)_{n\in\mathbb{N}}$ be finite or countable family of closed intervals with disjoint interiors and for any $I_{j}$ such that the set $I_{j}\cap\left(\left[0,1\right]\backslash\Omega^{\prime}\right)$ consists exactly of the endpoints of $I_{j}$.

Then there exists a probability measure $\mu$ absolutely continues with respect to the Lebesgue measure on $\left[0,1\right]$ such that $\left(\left[0,1\right],\mathcal{B},\mu,T,BV,\left\|\cdot\right\|_{BV}\right)$ fulfills Property $\mathfrak{C}$ if $T$ fulfills the following properties

• •

(Uniform expansion) $T_{n}\coloneqq T\lvert_{\mathring{I}_{n}}\in\mathcal{C}^{1}$, and $\left|T_{n}^{\prime}\right|\geq m>1$ for any $n\in\mathbb{N}$.

• •

$T$* is topologically mixing.*

• •

If we set $g\left(x\right)\coloneqq 1/\left|T^{\prime}\left(x\right)\right|$, then $g\lvert_{\mathring{I}_{i}}$ is a function of bounded variation for all $n\in\mathbb{N}$.

We note here that this example mainly relies on results in [Ryc83] on piecewise expanding interval maps on countable partitions generalizing [LY73] where finite partitions are considered.

Due to [Zwe98] the above properties are fulfilled for an infinite partition $\left(I_{n}\right)_{n\in\mathbb{N}}$ and the absolutely continuous measure $\mu$ is finite if

• •

(Adler’s condition) $T_{i}\coloneqq T\lvert_{\mathring{I}_{i}}\in\mathcal{C}^{2}$ and $T^{\prime\prime}/\left(T^{\prime}\right)^{2}$ is bounded on $\Omega^{\prime}$.

• •

(Finite image condition) $\#\left\{TI_{n}\colon I_{n}\in\mathcal{I}\right\}<\infty$.

• •

(Uniform expansion) $T_{n}\coloneqq T\lvert_{\mathring{I}_{n}}\in\mathcal{C}^{1}$, and $\left|T_{n}^{\prime}\right|\geq m>1$ for any $n\in\mathbb{N}$.

• •

$T$ is topologically mixing.

Proposition 1.12**.**

The system $\left(\left[0,1\right],\mathcal{B},\mu,T,BV,\left\|\cdot\right\|_{BV},\chi\right)$ with $\mu$ and $T$ from Proposition 1.11 fulfills Property $\mathfrak{D}$ with $\chi:\left[0,1\right]\to\mathbb{R}_{\geq 0}$ if there exist $\widetilde{K}_{1},\widetilde{K}_{2}>0$ such that for all $\ell\in\mathbb{R}_{\geq 0}$

[TABLE]

and

[TABLE]

Setting for example $\chi$ monotonically increasing or decreasing implies that (10) and (11) are fulfilled. Giving then a system $\left(\left[0,1\right],\mathcal{B},\mu,T,BV,\left\|\cdot\right\|_{BV}\right)$ which fulfills Property $\mathfrak{C}$ enables us to apply the theorems from the previous two sections.

Remark 1.13*.*

We note here that Property $\mathfrak{D}$ includes examples with a mixing structure as in [Hay14] not fulfilling a lightly trimmed strong law. Haynes investigated the dynamical system $\left(\left[0,1\right),\mathcal{B},\lambda_{[0,1)},T\right)$ with $Tx=2x\mod 1$ and the observable $\chi(x)=\left\lfloor 1/x\right\rfloor$ which does not fulfill a lightly trimmed strong law.

We might compare this system to i.i.d. random variables with the same distribution function as the above system and the $\bm{\psi}$-mixing continued fraction system mentioned in the introduction which also has the same distribution function with respect to the Lebesgue measure. Then we obtain that both systems fulfill a lightly trimmed strong law after only removing the maximal term. The i.i.d. case follows after an easy calculation using the results in [Mor77].

However, if we take the same system $\left(\left[0,1\right),\mathcal{B},\lambda_{[0,1)},T\right)$ but alter $\chi$ now being defined as $\chi(x)=\lfloor 1/x^{1/\alpha}\rfloor$ with $0<\alpha<1$, then the system $\left(\left[0,1\right],\mathcal{B},\lambda\lvert_{\left[0,1\right)},T,BV,\left\|\cdot\right\|_{BV},\chi\right)$ fulfills Property $\mathfrak{D}$, see the above propositions. In this case we have $\left(y+1\right)^{-\alpha}\leq\mu\left(\chi>x\right)\leq y^{-\alpha}$, i.e. $\mu\left(\chi>x\right)$ is regularly varying with index $-\alpha$. Applying Theorem 1.7 gives that for this system an intermediately trimmed strong law holds. By the subsequent Remark 1.9 the optimal trimming sequence $(b_{n})$ also does not change by the fact that the random variables are not independent.

So it seems that intermediately trimmed strong laws are less susceptible of dependence structures than lightly trimmed strong laws.

2. Main ideas of proofs

All our trimming theorems, i.e. Theorem 1.3 with its Corollary 1.5 and Theorem 1.7 for regularly varying tails follow the same idea of proof which will be given in the following. The remaining parts of the proof will then be given in Section 4.

2.1. Sketch of proof for almost sure limit theorems

In order to prove an intermediately trimmed strong law we use two main properties discussed in the following and denoted as Properties $\bm{A}$ and $\bm{B}$. We will first state the properties and then in Lemma 2.3 state how together with an additional condition they imply an intermediately trimmed strong law.

In the subsequent sections, i.e. Section 2.2 and Section 2.3 we will give an idea how to prove Properties $\bm{A}$ and $\bm{B}$, what distinguishes them from the case of i.i.d. random variables and why it is difficult to directly use results about mixing random variables instead of random variables fulfilling Property $\mathfrak{D}$, see Definition 1.2, i.e. fulfilling a spectral gap property.

The first property considers the sum of truncated random variables. Namely, for $\chi:\Omega\to\mathbb{R}_{\geq 0}$ we define

[TABLE]

for all $\ell\in\mathbb{R}_{\geq 0}$ and for a real valued sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$ we let

[TABLE]

denote the corresponding truncated sum process.

Definition 2.1**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$. We say that $(f_{n})$ fulfills Property $\bm{A}$ for the system $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ if

[TABLE]

The second property deals with the average number of large entries and is defined as follows.

Definition 2.2**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$. We say that a tuple $((f_{n}),(\gamma_{n}))$ fulfills Property $\bm{B}$ for the system $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ if

[TABLE]

In the following lemma we show how these two properties together with condition (13) give an intermediately trimmed strong law.

Lemma 2.3**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ which fulfill Property $\mathfrak{D}$. Further, let $(f_{n})$ fulfill Property $\bm{A}$ and let $((f_{n}),(\gamma_{n}))$ fulfill Property $\bm{B}$ for the system $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$. If additionally

[TABLE]

holds, then we have for $b_{n}\coloneqq\left\lceil n\cdot\mu\left(\chi>f_{n}\right)+\gamma_{n}\right\rceil$ that

[TABLE]

Proof.

We can conclude from Property $\bm{B}$ that a.s.

[TABLE]

On the other hand since $\prescript{f_{n}}{}{\mathrlap{\smash{\chi}}\phantom{\chi}}\leq f_{n}$ it follows by Property $\bm{B}$ that a.s.

[TABLE]

Combining (14) and (15) yields that a.s.

[TABLE]

Combining this with Property $\bm{A}$ and (13) gives the statement of the lemma. ∎

2.2. Statements about truncated random variables (Property A)

We will first state two results giving an answer for which sequences $(f_{n})$ Property $\bm{A}$ holds. Recall that we denote by $F$ the distribution function of $\chi$ with respect to $\mu$, i.e. $F(x)=\mu(\chi\leq x)$.

Theorem 2.4**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$. For a positive valued sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$ assume $F\left(f_{n}\right)>0$ for all $n\in\mathbb{N}$ and there exists $\psi\in\Psi$ such that

[TABLE]

holds. Then

[TABLE]

Theorem 2.5**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$. Assume that $F\left(x\right)=L\left(x\right)/x^{\alpha}$ where $0<\alpha<1$ and $L$ is slowly varying. Let $\left(f_{n}\right)_{n\in\mathbb{N}}$ be a positive valued sequence and assume that $F\left(f_{n}\right)>0$, for all $n\in\mathbb{N}$. If there exists $\psi\in\Psi$ such that

[TABLE]

then we have

[TABLE]

Remark 2.6*.*

If additionally $f_{n}$ tends to infinity, we have the more explicit statement that

[TABLE]

This can be easily obtained from Lemma 3.18.

Remark 2.7*.*

The above result also improves a result by Nakata for i.i.d. random variables, see [Nak15]. The results in [Nak15, Theorem 1.2 (ii)] are slightly more general in the sense that he considers two sequences $(l_{n})$ and $(c_{n})$ both depending on $n$, where $(c_{n})$ denotes the truncation, in our case denoted by $(f_{n})$ and $(l_{n})$ denotes the summation index which we alway set equal to $n$.

However, his results are restricted to the case that $1-F\left(x\right)\asymp x^{-\alpha}$, where $0<\alpha<1$ and $f(x)\asymp g(x)$ denotes the existence of $C>0$ such that $1/Cg(x)\leq f(x)\leq Cg(x)$, for all $x$.

One example for this setting is $F\left(x\right)=1-1/x^{\alpha}$ with $0<\alpha<1$. For this example his condition for (18) to hold can in our notation be written as

[TABLE]

Comparing this with the condition that there exists $\psi\in\Psi$ such that (17) holds with $L$ constant, shows that (17) is indeed a weaker condition.

2.3. A statement about a large deviation result (Property B)

The following lemma gives us a statement to determine under which conditions Property $\bm{B}$ holds. The formulation is very general so that it enables us to later prove trimming statements for different settings.

Lemma 2.8**.**

Set

[TABLE]

for $k\in\mathbb{R}_{\geq 1}$, $n\in\mathbb{N}_{\geq 3}$, $0<\epsilon<1/4$, and $\psi\in\Psi$. There exists a constant $V>0$ such that for all $\epsilon\in(0,1/4)$, $\psi\in\Psi$, and all positive valued sequences $\left(u_{n}\right)_{n\in\mathbb{N}}$ and $p_{n}\coloneqq\mu\left(\chi>u_{n}\right)$, $n\in\mathbb{N}$, we have

[TABLE]

We note here that $\mathsf{S}_{n}\mathbbm{1}_{\left\{\chi>u_{n}\right\}}=\sum_{k=1}^{n}\mathbbm{1}_{\left\{\chi\circ T^{k-1}>u_{n}\right\}}$. The lemma is an analogous statement to [KS17, Theorem 8] for the setting of dynamical systems instead of independent random variables.

2.4. Comments about the method of proof

The method to first prove a statement as in Property $\bm{A}$ in order to prove an intermediately trimmed strong law coincides with the method used for many trimming statements in the i.i.d. setting, for instance in [HM87, Last part of the proof of Lemma 2] and [Mor77, Lemma 6].

The method to prove trimming results with a large deviation statement as in Property $\bm{B}$ is the same as in [Mor77] and [KS17], but differs for example from [HM87] who used a quantile transformation. However, the quantile transformation heavily relies on the independence of the random variables, so it does not seem reasonable to transfer this method to our case.

The main idea for proving that $\left(f_{n}\right)$ fulfills Property $\bm{A}$ is to use one version of an exponential inequality, in the i.i.d. case usually the Bernstein inequality, to estimate

[TABLE]

Using in the following a Borel-Cantelli argument for the right choice of $(f_{n})$ gives a statement as in (12). Similarly, to prove that a tuple $((f_{n}),(\gamma_{n}))$ fulfills Property $\bm{B}$ one can use an exponential inequality estimating

[TABLE]

in order to apply a Borel-Cantelli statement afterwards.

In particular for proving a Property $\bm{A}$ statement it is difficult to use an exponential inequality for mixing random variables. To explain this we will in the following distinguish the notation between random variables from a dynamical system, writing $\mathsf{S}_{n}\chi$, $\mathsf{T}^{r}_{n}\chi$, $\mu$, $(\chi\circ T^{n-1})$ and $\int\chi\mathrm{d}\mu$ and generic random variables, for example i.i.d. writing $S_{n}$, $T_{n}^{r}$, $\mathbb{P}$, $(X_{n})$, and $\mathbb{E}\left(X_{1}\right)$.

The first difficulty is that exponential inequalities usually depend on the maximal bound of the considered random variables. In our case we have that $\left|\prescript{f_{n}}{}{\chi}\right|_{\infty}=f_{n}$ and $f_{n}$ is depending on $n$. Using a direct application of the Bernstein inequality as in [KS17, Lemma 10] we obtain for the i.i.d. case that

[TABLE]

The easiest way to prove a statement as in (12) is then to determine $\left(f_{n}\right)$ such that

[TABLE]

In order to be precise we have to remark here that in our proof we actually use a different sum given in (64). However, the problem explained in the following does not change by considering the sum in (23) instead.

In case that we have mixing random variables the situation becomes more difficult than for independent random variables. Even though there is a vast literature about exponential inequalities for mixing random variables, it is difficult to obtain results as strong as in the i.i.d. case using them. We will explain the difficulties by explaining some of the more recent results. Many exponential inequalities are given with a constant depending on the distribution function of the considered random variable and, consequently, depend on the bounds $\left(f_{n}\right)$ and the dependence can not be easily determined by the proof of these inequalities, this is for example the case in [MPR11] and some statements in [Dou94]. For instance, [MPR11, Theorem 1] states that under some regularity conditions on the random variables $(X_{n})$ there exist constants $\gamma$ and $C_{1}$ such that

[TABLE]

In their theorem, $\gamma$ can be calculated by the distribution function of $X_{1}$ and the mixing properties of $(X_{n})$, but $C_{1}$ is dependent on $\gamma$ which changes if one considers $S_{n}^{f_{n}}$ with $f_{n}$ not constant.

Other results, for example [Ada08], require a more restricted setting than considered here, covering only Markov chains.

In other publications as in [MPR09] the bound depending on $\left(f_{n}\right)$ is given, but the obtained results are weaker than in the i.i.d. case. Theorem 2 of this publication states that for exponentially $\bm{\alpha}$-mixing identically distributed random variables $(X_{n})$ there exists a constant $C>0$ such that for all $n\in\mathbb{N}$ and $x\geq 0$

[TABLE]

where $\left|X_{i}\right|\leq M$ and $v^{2}=\sup_{i>0}\left(\mathbb{V}\left(X_{i}\right)+2\sum_{j>i}\left|\mathrm{Cov}\left(X_{i},X_{j}\right)\right|\right)>0$ and $\mathbb{V}(Y)$ denotes the variance of $Y$. Applying this to our case this result yields

[TABLE]

For the example $F\left(x\right)=1-1/x^{\alpha}$ with $0<\alpha<1$ we have by (43) that

[TABLE]

As a comparison, in the i.i.d. case we obtain by the Bernstein inequality (22) for the same distribution function

[TABLE]

This difference in the denominator changes the set of sequences $(f_{n})$ which fulfill (23) a lot and thus also statements about the set of sequences $(f_{n})$ fulfilling Property $\bm{A}$. Similar problems occur by applying other theorems in this publication.

Other statements, see for example [Dou94, Theorem 4, Chapter 1.4.2], give an exponential inequality with an additional summand depending on the speed of mixing. For $\left(X_{n}\right)_{n\in\mathbb{N}}$ being a sequence of non-negative, $\bm{\beta}$-mixing (a slightly weaker form of mixing than $\bm{\phi}_{\textrm{rev}}$-mixing), identically distributed random variables with $\lim_{n\rightarrow\infty}\bm{\beta}\left(n\right)\cdot n=0$ Doukhan states for any $0<\delta<1$ and $0\leq\eta\leq n/\left(1+\delta^{2}/4\right)$ that

[TABLE]

where $\left|X_{1}\right|\leq M$ and $\sigma^{2}\geq\mathbb{V}\left(\sum_{k=i}^{i+j-1}Y_{k}\right)/j$ for all $i,j\in\mathbb{N}$.

In order to obtain a summable expression for an estimate of (21) one would have to guarantee that both summands in (24) are summable. For the second summand this can only be ensured if $\eta_{n}\coloneqq\eta$ is tending to infinity for $n$ tending to infinity. However, this increases the first summand, leading to worse results than in the i.i.d. case.

Further, we want to point out that Adamczak gives an example in [Ada08] that for observables on an infinite state aperiodic and irreducible Markov chain the estimation can not be as good as in the independent case in that sense that the expression in the exponent differs by more than only a constant. Even though it is not clear if this example fulfills a $\bm{\phi}_{\textrm{rev}}$-mixing property and it does not fulfill a $\bm{\phi}$-mixing property, by [Bra05, Theorem 3.2] this Markov chain is at least exponentially $\bm{\beta}$-mixing. From this consideration it might be reasonable to assume that the lack of independence is a reason that exponential inequalities for $\bm{\phi}_{\textrm{rev}}$-mixing random variables can not be formulated with the same sharpness as in the i.i.d. case.

Instead of using results for mixing random variables we prove an exponential inequality for dynamical systems fulfilling Property $\mathfrak{C}$, stated as follows:

Lemma 2.9**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$. Then there exist positive constants $K$, $N$, $U$ such that for all $\varphi\in\mathcal{F}$ fulfilling $\int\varphi\mathrm{d}\mu=0$, all $u\in\mathbb{R}_{>0}$, and all $n\in\mathbb{N}_{>N}$ we have

[TABLE]

Additionally to their mixing properties the random variables $\left(\varphi\circ T^{n-1}\right)$ obtain additional structures which facilitate to obtain an exponential inequality. More precisely, we first use Markov’s inequality to obtain the statement in Lemma 4.2. The subsequent estimation of $\int\exp\left(t\cdot\mathsf{S}_{n}\varphi\right)\mathrm{d}\mu$ seems to give better results using analytic perturbation theory than using blocking techniques for mixing random variables.

The main idea here is to use perturbation theory for the perturbed transfer operator $\widehat{T}_{z}h\coloneqq\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}h\coloneqq\widehat{T}\left(\mathrm{e}^{z\cdot\varphi}\cdot h\right)$. This gives us $\int\mathrm{e}^{t\cdot\mathsf{S}_{n}\varphi}\mathrm{d}\mu=\int\widehat{T}_{t}^{n}1\mathrm{d}\mu$, see the proof of Lemma 4.3. We assume that $\widehat{T}$ has a spectral gap and prove in Lemma 3.5 that this is also true for $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{t}}}\phantom{T_{t}}}$ if $\left\|\varphi\right\|\cdot t$ is small enough. This will eventually lead us to the result that $\int\widehat{T}_{t}^{n}1\mathrm{d}\mu\leq K^{\prime}\cdot\left|\prescript{\varphi}{}{\mathrlap{\smash{\lambda}}\phantom{\lambda}}_{t}\right|^{n}$ with $\lambda_{t}\coloneqq\prescript{\varphi}{}{\mathrlap{\smash{\lambda}}\phantom{\lambda}}_{t}$ being the largest eigenvalue of the operator $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}}}\phantom{T}}_{t}$ and $K^{\prime}$ a constant, see Lemma 4.3. Finally, we estimate $\lambda_{t}$ using the Taylor expansion $\sum_{k=0}^{\infty}t^{k}\cdot\lambda_{0}^{\left(k\right)}/k!$, see Lemma 4.4.

The proof of Lemma 2.9 seems rather lengthy compared to the proof of limit theorems using analytic perturbation theory, for example the central limit theorem. One reason for that is that in Lemma 3.5 we need a uniform bound of $\left\|\varphi\right\|\cdot t$ for $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{t}}}\phantom{T_{t}}}$ to fulfill a spectral gap property. Such results are usually only developed for $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{t}}}\phantom{T_{t}}}$ for a fixed $\varphi$. A similar problem occurs in Lemma 3.7.

Furthermore, in order to obtain optimal results we indeed have to estimate $\lambda_{0}^{(k)}$, for all $k\in\mathbb{N}$, see Lemma 4.4, not stopping after a certain term of the Taylor expansion.

Additionally, we prove this exponential inequality as a maximal inequality. Even though there is literature generalizing exponential inequalities to maximal exponential inequalities, see [KM13], we obtain better results using a direct approach, i.e. we transform the sum $\mathsf{S}_{n}\varphi$ into a martingale and apply a generalization of Doob’s inequality, see Lemma 4.1.

With respect to Lemma 2.8 we remark that the considered random variables are all bounded by $1$ and it might therefore be possible to directly use exponential inequalities stated for mixing random variables. However, for brevity we used Lemma 2.9 again. The obtained results only differ from the results for i.i.d. random variables in the respect that in the i.i.d. case the constant $V$ in Lemma 2.8 can be given explicitly, whereas in our case the constant depends on some inherent properties of the dynamical system.

The second difficulty is that in order to prove optimal results we use some inclusions in the proof of Theorem 2.5 as well as in the proof of Lemma 2.8, i.e. we consider in fact a more complicated term than (21). The method is mentioned as (A) and (B) on p. (A) for the proof of Theorem 2.5 and as (A) and (B) on p. (A) for the proof of Lemma 2.8. This enables us to use the Borel-Cantelli lemma for an exponential subsequence leading to stronger results.

The method used in [HM87] is slightly shorter, however it requires $\left(f_{n}\right)$ to be additionally monotonic. With this result it is only possible to prove trimming statements with $\left(b_{n}\right)$ being asymptotic to an increasing sequence. Sequences $\left(b_{n}\right)$ as considered in Remark 1.9 are not covered in their results, since following the proof of Theorem 1.7 gives $b_{n}\sim n\cdot\mu\left(\chi>f_{n}\right)$. Thus, $f_{n}$ being monotonic does not allow for large oscillations of $b_{n}$.

2.5. Structure of the paper

We will give some preliminary statements in Section 3, i.e. we will give the definition of a spectral gap and show the resulting decay of correlation in Section 3.1. In Section 3.2 we give an introduction into analytic perturbation theory for the operator $\widehat{T}_{z}$ defined as $\widehat{T}_{z}h\coloneqq\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}h\coloneqq\widehat{T}\left(\mathrm{e}^{z\cdot\varphi}\cdot h\right)$. We will improve two classical results by showing that they also hold uniformly. In Section 3.3 we will recall some classical results in the theory of regular variation which enables us later to prove Theorem 1.7.

With this background we are able to prove the main steps explicated in the previous section. In Section 4.1 we prove the exponential inequality Lemma 2.9. Using this inequality we can prove the statements concerning Properties $\bm{A}$ and $\bm{B}$, i.e. in Section 4.2 we prove Theorems 2.4 and 2.5 and in Section 4.3 we prove Lemma 2.8. Finally, in Section 4 we prove our main theorems, i.e. we prove the necessary properties to apply Lemma 2.3 and show then how the trimming statements in Sections 1.2 and 1.3 follow from this lemma. In the last Section 4.5 we prove that our example from Section 1.4 actually fulfills all required properties.

3. Preliminary statements

3.1. Spectral gap and decay of correlation

In this section we will first give the precise definition of a spectral gap. Then we state some properties for dynamical systems fulfilling Property $\mathfrak{C}$, for instance a decay of correlations.

The results of this section are known, but scattered around the literature and often only proven for one particular system why we decided to reformulate them in the general setting.

In the following we denote the spectral radius of an operator $U$ by $\rho\left(U\right)$. The following definition is a key part of Property $\mathfrak{C}$ for $\widehat{T}$, see Definition 1.1.

Definition 3.1** (Spectral gap).**

Suppose $\mathcal{F}$ is a Banach space and $U:\mathcal{F}\to\mathcal{F}$ a bounded linear operator. We say that $U$ has a spectral gap if there exists a decomposition $U=\lambda P+N$ with $\lambda\in\mathbb{C}$ and $P,N$ bounded linear operators such that

• •

$P$ is a projection, i.e. $P^{2}=P$ and $\dim\left(\operatorname{Im}\left(P\right)\right)=1$,

• •

$N$ is such that $\rho\left(N\right)<\left|\lambda\right|$,

• •

$P$ and $N$ are orthogonal, i.e. $PN=NP=0$.

Dealing with operators $U:\mathcal{F}\to\mathcal{F}$ we use the operator norm $\left\|U\right\|\coloneqq\sup_{\left\|\varphi\right\|\leq 1}\left\|U\varphi\right\|$.

Lemma 3.2**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$. Then $\widehat{T}$ has a simple eigenvalue $\lambda=\rho\left(\widehat{T}\right)=1$. This eigenvalue is unique on the unit circle and has maximal modulus.

Proof.

Using the defining relation of $\widehat{T}$ in (1) gives that its operator norm with respect to the $\mathcal{L}_{\mu}^{1}$-norm is equal to $1$. This immediately implies that the modulus of any eigenvalue cannot exceed $1$. Now, if $\lambda$ is an eigenvalue of the eigenfunction $f$, then we have for every $A\in\mathcal{B}$ and $n\in\mathbb{N}$ that

[TABLE]

If we assume that $\lambda=1$, then for all $a\in\mathbb{R}$ and $n\in\mathbb{N}$, we have

[TABLE]

Since $T$ is ergodic and preserves the probability measure $\mu$ it follows that $\mu(\left\{f>a\right\})\in\{0,1\}$, for all $a\in\mathbb{R}$, and consequently $f$ has to be constant almost everywhere. Since $1$ is the eigenvalue only for the constant functions, this eigenvalue has to be simple. Furthermore, since $\mu$ is mixing we have for all $f\in\mathcal{L}^{2}$ and thus $f\in\mathcal{F}$ and all $A\in\mathcal{B}$ that

[TABLE]

If we assume that $f$ is an eigenfunction to the eigenvalue $\lambda$, (25) implies

[TABLE]

In case that $\lambda$ lies on the unit circle the only possibility that this equality holds is that $\lambda=1$. ∎

Lemma 3.3**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$ and write $\widehat{T}=\lambda P+N$ with $\lambda,P,N$ as in Definition 3.1. Then $Pf=\int f\mathrm{d}\mu$ holds for all $f\in\mathcal{F}$.

Proof.

From the proof of Lemma 3.2 we know that the constant functions are eigenfunctions of $\widehat{T}$ to the maximal eigenvalue $\lambda=1$. Thus, $Pf$ is almost everywhere constant for all $f\in\mathcal{F}$. By Riesz’ representation theorem we know that there exists $g\in\mathcal{L}^{\infty}$ such that for all $f\in\mathcal{F}$ we have that $Pf=\int f\cdot g\mathrm{d}\mu$. Furthermore,

[TABLE]

Since $\mu$ is ergodic, $g=g\circ T$ and thus, $g$ is constant. ∎

With the above lemmas we can prove that a decay of correlation holds under Property $\mathfrak{C}$.

Lemma 3.4** (Decay of correlations).**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$. Further, let $\varphi_{1}\in\mathcal{L}^{1}$ and $\varphi_{2}\in\mathcal{F}$ and define

[TABLE]

Then there exists for every $\tau>\overline{\tau}$ a constant $R>0$ such that

[TABLE]

for all $n\in\mathbb{N}$.

The proof of this lemma follows the proof of [Bal00, Theorem 1.6] which is given for piecewise expanding interval maps.

Proof.

By the properties of the transfer operator we have that

[TABLE]

It follows from Lemmas 3.2 and 3.3 that \operatorname{spec}\big{(}\widehat{T}\big{)} consists of the simple eigenvalue $1$ with the corresponding spectral projection $Pf=\int f\mathrm{d}\mu$ and a subset of a disc with radius $\overline{\tau}<1$. Hence,

[TABLE]

and it follows from Gelfand’s formula, see for example [Lax02, p. 195], and $\left|\cdot\right|_{\infty}\leq\left\|\cdot\right\|$ that for all $\tau>\overline{\tau}$ there exists $R>0$ such that for all $n\in\mathbb{N}$

[TABLE]

and thus, the statement of the lemma follows. ∎

3.2. Analytic perturbation of the transfer operator

We consider in the following the perturbed transfer operator given by

[TABLE]

Obviously, $\widehat{T}=\widehat{T}_{0}$. We remember from the previous section that under the restriction that condition $\mathfrak{C}$ holds we have that $\widehat{T}=P+N\coloneqq P_{0}+N_{0}$ with $\rho(N_{0})<1$.

Our main goal in this section is to prove the following two main lemmas, where the first reads as follows.

Lemma 3.5**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$. For all $\theta\in\left(0,(1-\rho(N_{0}))/2\right)$ there exists $\varkappa>0$ such that for all $\varphi\in\mathcal{F}$ with $\int\varphi\mathrm{d}\mu=0$ and $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa$ we have

[TABLE]

where

[TABLE]

and $T_{z}$, $\lambda_{z}$, $P_{z}$, and $N_{z}$ are analytic with the definition of analyticity for operators given in Definition 3.6.

The fact that for a given function $\varphi$ the operator $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}$ is analytic and has a decomposition $\prescript{\varphi}{}{\mathrlap{\smash{\lambda_{z}}}\phantom{\lambda_{z}}}\cdot\prescript{\varphi}{}{\mathrlap{\smash{P_{z}}}\phantom{P_{z}}}+\prescript{\varphi}{}{\mathrlap{\smash{N_{z}}}\phantom{N_{z}}}$, where $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}$, $\prescript{\varphi}{}{\mathrlap{\smash{\lambda_{z}}}\phantom{\lambda_{z}}}$, $\prescript{\varphi}{}{\mathrlap{\smash{P_{z}}}\phantom{P_{z}}}$, and $\prescript{\varphi}{}{\mathrlap{\smash{N_{z}}}\phantom{N_{z}}}$ are analytic in a neighborhood of zero is widely known, see for example [Gou15, Proposition 2.3], [FJ03, Theorem 8.2], [Sar12, Chapter 3].

However, those results say nothing about a uniform bound of the analytic convergence radius if one considers the perturbed operator $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}$ for different functions $\varphi$. Furthermore, to prove Lemma 3.7 we will see that the additional uniform bounds $\left|\lambda_{z}\right|\geq 1-\theta$ and $\rho(N_{z})\leq\rho(N_{0})+\theta$ instead of just assuming $\rho(N_{z})<\lambda_{z}$ are necessary.

When we speak here about analyticity of operators we usually refer to strong analyticity given in the following definition.

Definition 3.6** (Weak and strong analyticity).**

Let $\mathcal{G}$ be a Banach space and $B\coloneqq B\left(\mathcal{G}\right)$ be the space of all bounded linear operators. Let $U\subset\mathbb{C}$ be open and $\left(L_{z}\right)_{z\in U}$ a one parameter family of operators in $B$.

We call $\left(L_{z}\right)$ weakly analytic if $\varphi\left(L_{z}\right)$ is homomorphic on $U$ for every bounded linear $\varphi:B\to\mathbb{C}$.

We call $\left(L_{z}\right)$ strongly analytic if for every $z\in U$ there exists $L^{\prime}_{z}\in U$ such that

[TABLE]

However, from [Kat95, Theorem III-3.12] we know that in fact weak analyticity and strong analyticity are equivalent.

Our second main lemma in this section will give a bound on $n$ such that $\left|\prescript{\varphi}{}{\mathrlap{\smash{\lambda}}\phantom{\lambda}}_{z}\right|^{-n}\cdot\left\|\prescript{\varphi}{}{\mathrlap{\smash{N}}\phantom{N}}_{z}^{n}\right\|\leq 1$.

Lemma 3.7**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$ and let $\prescript{\varphi}{}{\mathrlap{\smash{\lambda}}\phantom{\lambda}}_{z}$, $\prescript{\varphi}{}{\mathrlap{\smash{N}}\phantom{N}}_{z}$, and $\varkappa$ be given as in Lemma 3.5. Then there exists $N\in\mathbb{N}$ and $\varkappa^{\prime}\in\left(0,\varkappa\right)$ such that for all $\varphi\in\mathcal{F}$ fulfilling $\int\varphi\mathrm{d}\mu=0$, all $z\in\mathbb{C}$ with $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa^{\prime}$, and all $n\in\mathbb{N}_{>N}$

[TABLE]

From Gelfand’s formula, see for example [Lax02, p. 195], and Lemma 3.5 it follows easily that

[TABLE]

and for fixed $\varphi$ an $N$ fulfilling (29) for all $n\geq N$ is easily determined. However, as we will see in the proof of this lemma at the end of the section, some more attention is needed to obtain a uniform bound.

In what follows we will recall some of the known results and prove with general operator theory the above lemmas. In the following let $\varphi_{z}^{\left(k\right)}$ with $z\in U$ and $U$ an open domain in $\mathbb{C}$, denote the $k$-th derivative of operators or $\mathbb{C}$ valued functions $\left(z\mapsto\varphi_{z}\right)_{z\in U}$.

Lemma 3.8**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$ and let $\varphi\in\mathcal{F}$. Then the operator $\widehat{T}_{z}$ given by (26) is analytic on $\mathbb{C}$ and we have

[TABLE]

where $M_{\varphi}:\mathcal{F}\to\mathcal{F}$ and $M_{\varphi}h\coloneqq\varphi h$.

Proof.

To show the analyticity of $\widehat{T}_{z}$, observe that by definition

[TABLE]

in a neighborhood of zero. $M_{\varphi}$ is bounded, because $\left\|M_{\varphi}h\right\|\leq\left\|\varphi\right\|\left\|h\right\|$. Therefore,

[TABLE]

converges, i.e. $\widehat{T}_{z}$ converges absolutely with respect to $\left\|\cdot\right\|$ and the representation in (31) holds for all $z\in\mathbb{C}$. Hence, for each $\varphi$ element of the dual of the space of bounded linear operators, \varphi\,\big{(}\widehat{T}_{z}\big{)} can be expanded into a power series. Since strong and weak analyticity are equivalent it follows that $\widehat{T}_{z}^{\prime}$ and by the analyticity also the derivatives $\widehat{T}_{z}^{\left(n\right)}$ exist. (30) follows from the series expansion (31) by induction. ∎

In the following we are considering for which $\varphi$ and $z\in\mathbb{C}$ the operator $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}}}\phantom{T}}_{z}$ is close to $\widehat{T}$. In order to specify closeness we start with some definitions concerning the norms.

Definition 3.9**.**

For $M\subset\mathcal{F}$ set $S_{M}\coloneqq\left\{u\in M\colon\left\|u\right\|=1\right\}$. We define for two sets $M,N\subset\mathcal{F}$ and $u\in\mathcal{F}$ the distance measures

[TABLE]

Furthermore, we define for two operators $U,V:\mathcal{F}\to\mathcal{F}$

[TABLE]

where $G\left(U\right)$ denotes the graph of $U$. As the graph norm for a graph of an operator $U:\mathcal{F}\to\mathcal{F}$ we use for $\left(x,y\right)\in\mathcal{F}\times\mathcal{F}$ the norm

[TABLE]

Remark 3.10*.*

Note that in [Kat95] $d$ and $\widehat{d}$ are denoted by $\delta$ and $\widehat{\delta}$, whereas $d$ are $\widehat{d}$ are used in there to denote some other distance measures.

Indeed, the convergence with respect to $\widehat{d}$ is a generalization to convergence in norm as the following calculation for $U,V\colon\mathcal{F}\to\mathcal{F}$ shows: By Definition 3.9 it follows that

[TABLE]

Similarly, we obtain

[TABLE]

Combining (32) and (33) yields

[TABLE]

Lemma 3.11**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$. Let $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}$ be defined as in (26). Then for all $\epsilon>0$ there exists $\delta>0$ such that $\left|z\right|\cdot\left\|\varphi\right\|<\delta$ implies $\widehat{d}\left(\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}},T\right)<\epsilon$.

Proof.

Let $\epsilon>0$ be given such that \widehat{d}\,\big{(}\widehat{T},\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}\big{)}<\epsilon. By the definition of $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}$ and inequalities (3) and (34) we obtain

[TABLE]

where the last inequality follows from the fact that $\mathcal{F}$ is a Banach algebra. Furthermore, we have that

[TABLE]

and we can thus conclude from (35) that

[TABLE]

Furthermore, there exists $\delta>0$ such that $\left\|\varphi\right\|\cdot\left|z\right|<\delta$ implies $K_{0}\cdot\left(\exp\left(\left\|\varphi\right\|\cdot\left|z\right|\right)-1\right)<\epsilon$, which gives the statement of the lemma. ∎

In the next steps we prove that the operator $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}$, if close enough to $\widehat{T}$, can indeed be spectrally decomposed similarly as $\widehat{T}$ without saying anything about the analyticity of the spectral components. More precisely we formulate this in the following lemma.

Lemma 3.12**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$. Let $\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}$ be defined as in (26). Then there exists $\kappa>0$ such that for all $\varphi\in\mathcal{F}$ with $\int\varphi\mathrm{d}\mu=0$ and $\left|z\right|\cdot\left\|\varphi\right\|<\kappa$ a spectral decomposition as in (27) exists fulfilling (28).

For the proof of Lemma 3.12 we will basically use the following parts of [Kat95, Theorem IV-3.16]:

Lemma 3.13**.**

Let $\mathcal{G}$ be a Banach space, let $U:\mathcal{G}\to\mathcal{G}$ be a bounded linear operator, and let $\Sigma\left(U\right)\coloneqq\operatorname{spec}\left(U\right)$ be separated into two parts $\Sigma^{\prime}\left(U\right)$ and $\Sigma^{\prime\prime}\left(U\right)$ by a simple closed curve $\Gamma$. Let $\mathcal{G}=M^{\prime}\left(U\right)\oplus M^{\prime\prime}\left(U\right)$ be the associated decomposition of $\mathcal{G}$. Then there exists $\epsilon>0$ depending on $U$ and $\Gamma$ such that for all $V:\mathcal{G}\to\mathcal{G}$ with $\widehat{d}\left(U,V\right)<\epsilon$ the spectrum $\Sigma(V)\coloneqq\operatorname{spec}(V)$ is separated by $\Gamma$ into $\Sigma^{\prime}(V)$ and $\Sigma^{\prime\prime}(V)$ and the associated decomposition

[TABLE]

fulfills

[TABLE]

The decomposition $\mathcal{G}=M^{\prime}(V)\oplus M^{\prime\prime}(V)$ is continuous in $V$ in the sense that for all $\epsilon>0$ there exists $\delta>0$ such that $\widehat{d}\left(U,V\right)<\delta$ implies $\left\|P[U]-P[V]\right\|<\epsilon$, where $P[U]$ denotes the projection of $\mathcal{G}$ onto $M^{\prime}(V)$ along $M^{\prime\prime}(V)$ and $P[V]$ analogously.

The lemma holds analogously for separating the spectrum in more than one component.

Proof of Lemma 3.12.

Since $\widehat{T}$ has a spectral gap, we can write

[TABLE]

with $\lambda_{0}$, $P_{0}$, and $N_{0}$ fulfilling (28). For proving that $\widehat{T}_{z}$ can be represented in the same manner we aim to apply Lemma 3.13 splitting the spectrum formally in three components. First we choose for $\Gamma=\Gamma_{1}$ a circle around [math] with radius $r=\rho(N_{0})+\theta$ and as a second step we choose for $\Gamma=\Gamma_{2}$ a circle around [math] with radius $r=1-\theta$. By the fact that $\widehat{T}$ has a spectral gap and Lemma 3.2 this separates the spectrum of $\widehat{T}$ into the single eigenvalue $1$ (outside the circle $\Gamma_{1}$), an empty part of the spectrum (between the circles $\Gamma_{1}$ and $\Gamma_{2}$) and the remainder of the spectrum \operatorname{spec}\big{(}\widehat{T}\big{)}\backslash\left\{1\right\} (inside the circle $\Gamma_{2}$).

These circles determine $\epsilon$ such that the spectrum of all $U:\mathcal{F}\to\mathcal{F}$ with \widehat{d}\,\big{(}\widehat{T},U\big{)}<\epsilon is separated in the above described way. Hence, \widehat{d}\,\big{(}\widehat{T},\prescript{\varphi}{}{\mathrlap{\smash{\widehat{T}_{z}}}\phantom{T_{z}}}\big{)}<\epsilon implies that a spectral decomposition as in (27) exists fulfilling (28). Using Lemma 3.11 and choosing $\kappa$ appropriately gives the statement of the lemma. ∎

Finally, to prove Lemma 3.5 we will use [Kat95, Theorems VII-1.7 and VII-1.8] and their proofs. Together they state the following.

Lemma 3.14**.**

If a family of operators $\left(U_{z}\right)_{z\in\mathbb{C}}$ with $U_{z}:\mathcal{F}\to\mathcal{F}$ depends on $z$ holomorphically and has a spectrum consisting of two separated parts and each $U_{z}$ can be represented as $U_{z}=\lambda_{z}P_{z}+N_{z}$ fulfilling the requirements as in (28), then there exists $\epsilon>0$ such that $\lambda_{z}$, $P_{z}$, and $N_{z}$ consist of branches of one or several analytic functions with at most algebraic singularities in $B\left(\epsilon,0\right)$, i.e. the ball with radius $\epsilon$ around zero. Let

[TABLE]

and

[TABLE]

then $\epsilon$ can be determined as the supremum of $r\in\mathbb{R}_{>0}$ such that (37) and (38) hold for all $\left|z\right|\leq r$ and all $f\in\mathcal{F}$.

Proof of Lemma 3.5.

By Lemma 3.12 there exists $\kappa>0$ such that the representation $\widehat{T}_{z}=\lambda_{z}P_{z}+N_{z}$ fulfilling (28) exists if $\left|z\right|\cdot\left\|\varphi\right\|<\kappa$. We aim to apply Lemma 3.14 on $\widehat{T}_{z}$. The analyticity of $\widehat{T}_{z}$ is given by Lemma 3.8. By (3) we have that

[TABLE]

and by (36) it follows that

[TABLE]

Hence, $U_{z}\coloneqq\widehat{T}_{z}$ fulfills (37) if we set $a\coloneqq K_{0}\cdot\left(\exp\left(\left|z\right|\cdot\left\|\varphi\right\|\right)-1\right)$ and $b\coloneqq 0$. Since $\exp\left(\left|z\right|\cdot\left\|\varphi\right\|\right)-1$ can be chosen arbitrarily close to zero for sufficiently small $\left|z\right|$, there exists $\varkappa\in\left(0,\kappa\right]$ such that $a\leq 1/\|\widehat{T}_{0}^{-1}\|$ and thus (38) holds for all $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa$.

Since (27) holds and there is in particular only one branch in $z=0$, it follows that there are no algebraic singularities in $B\left(\epsilon,0\right)$ and $\lambda_{z}$ is analytic on $B\left(\epsilon,0\right)$. ∎

Finally, we want to prove Lemma 3.7. First we define for a bounded operator $U$ the resolvent $R\left(\xi,U\right)\coloneqq(\xi\mathrm{id}-U)^{-1}$ and the resolvent set of $U$ as the set of all scalars $\xi$ such that $R\left(\xi,U\right)<\infty$. Before we start with the proof of this lemma, we first state the following lemma about the perturbation of the resolvent which is a simplified version of [Kat95, Theorem IV-3.15].

Lemma 3.15**.**

Let $\mathcal{G}$ be a Banach space and $U\colon\mathcal{G}\to\mathcal{G}$ be a bounded linear operator. For all $\epsilon>0$ there exists $\delta>0$ such that for all bounded operators $V\colon\mathcal{G}\to\mathcal{G}$ we have that $\widehat{d}(U,V)<\delta$ implies $\left\|R(\xi,V)-R(\xi,U)\right\|<\epsilon$.

Proof of Lemma 3.7.

Following the proof of Gelfand’s formula, see for example [Lax02, p. 196], yields for all $\delta>0$ and $n\in\mathbb{N}$ that

[TABLE]

with

[TABLE]

In the next steps we will show that there exists $\varkappa^{\prime}\in\left(0,\varkappa\right)$ such that $\prescript{\varphi}{}{\mathrlap{\smash{c}}\phantom{c}}_{z}$ is uniformly bounded for $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa^{\prime}$. Applying Lemma 3.5 with $\theta\coloneqq\left(1-\rho\left(N_{0}\right)\right)/4$ yields $\sup_{\left|z\right|<\varkappa/\left\|\varphi\right\|}\rho\left(N_{z}\right)\leq\rho\left(N_{0}\right)+\theta$ if $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa$. Thus, the maximum principle and choosing $\delta\coloneqq\theta$ yield for $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa$ that

[TABLE]

Our choice of $\theta$ ensures that $\zeta$ belongs to the resolvent set of $\prescript{\varphi}{}{\mathrlap{\smash{N}}\phantom{N}}_{z}$ if $\left|\xi\right|=\rho\left(N_{0}\right)+2\theta$.

We note here that (34) implies $\left\|N_{z}-N_{0}\right\|\geq\widehat{d}\left(\prescript{\varphi}{}{\mathrlap{\smash{N}}\phantom{N}}_{z},N_{0}\right)$. In the following we combine Lemma 3.11, the continuity statement of Lemma 3.13 applied for the projection to $N$, and Lemma 3.15. This implies that for all $\xi$ with $\left|\xi\right|=\rho\left(N_{0}\right)+2\theta$ there exists $\varkappa_{\xi}^{\prime}>0$ such that $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa_{\xi}^{\prime}$ implies $\left\|R\left(\xi,\prescript{\varphi}{}{\mathrlap{\smash{N}}\phantom{N}}_{z}\right)-R\left(\xi,N_{0}\right)\right\|\leq 1$.

Since the set $\left\{\xi\colon\left|\xi\right|=\rho\left(N_{0}\right)+2\theta\right\}$ is compact, we can set $\varkappa^{\prime}\coloneqq\min_{\left|\xi\right|=\rho\left(N_{0}\right)+2\theta}\varkappa_{\xi}^{\prime}>0$. Hence,

[TABLE]

Before we estimate $\left|\prescript{\varphi}{}{\mathrlap{\smash{\lambda}}\phantom{\lambda}}_{z}\right|^{-n}\cdot\left\|\prescript{\varphi}{}{\mathrlap{\smash{N}}\phantom{N}}_{z}^{n}\right\|$ we remember that Lemma 3.5 gives $\inf_{\left|z\right|<\varkappa/\left\|\varphi\right\|}\left|\prescript{\varphi}{}{\mathrlap{\smash{\lambda}}\phantom{\lambda}}_{z}\right|\geq 1-\theta$ and $\sup_{\left|z\right|<\varkappa/\left\|\varphi\right\|}\rho\left(\prescript{\varphi}{}{\mathrlap{\smash{N}}\phantom{N}}_{z}^{n}\right)\leq\rho\left(N_{0}\right)+\theta$. Applying this and the choice of $\delta=\theta$ on (39) yields for all $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa^{\prime}$ that

[TABLE]

Since by construction $\rho\left(N_{0}\right)+2\theta<1-\theta$, there exists $N\in\mathbb{N}$ such that for all $n>N$

[TABLE]

and thus $\left|\lambda_{z}\right|^{-n}\cdot\left\|N_{z}^{n}\right\|<1$, i.e. the statement of the lemma. ∎

3.3. Regular variation

In this section we give the definition of the de Bruijn conjugate, needed in Theorem 1.7 and some technical lemmas used for the regular variation setting. We start with the de Bruijn conjugate.

Definition 3.16**.**

Let $L$ be a slowly varying function at infinity. If the function $L^{\#}$ is slowly varying at infinity and fulfills the following convergences

[TABLE]

it is called the de Bruijn conjugate of $L$.

The de Bruijn conjugate always exists and is unique up to asymptotic equivalence. For further information see [BGT87, Section 1.5.7 and Appendix 5]. With the notion of the de Bruijn conjugate we are able to give asymptotic inverses for regularly varying functions given in the next two lemmas.

Lemma 3.17**.**

Let $\gamma,\delta>0$. If $f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is such that $f\left(x\right)=x^{\gamma\cdot\delta}L^{\gamma}\left(x^{\delta}\right)$, where $L$ denotes a slowly varying function in infinity, then any function $g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ with

[TABLE]

is the asymptotic inverse of $f$, i.e. $f\left(g\left(x\right)\right)\sim g\left(f\left(x\right)\right)\sim x$ for $x$ tending to infinity. One version of $g$ is $f^{\leftarrow}$.

Proof.

[BGT87, Proposition 1.5.15] states that (40) is the asymptotic inverse of $f$. Proposition 1.5.12 of the same book states that $h$ with $h\left(y\right)\coloneqq\inf\left\{x\in\left[0,\infty\right)\colon f\left(x\right)>y\right\}$ is one version of the asymptotic inverse and the asymptotic inverse is unique up to asymptotic equivalence. So we are left to show that $h\left(y\right)\sim f^{\leftarrow}\left(y\right)$. We assume the contrary of the statement. Obviously, $f^{\leftarrow}\leq h$. Then there exists $\epsilon>0$ such that $f^{\leftarrow}\left(y\right)\left(1+\epsilon\right)\leq\inf\left\{x\in\left[0,\infty\right)\colon f\left(x\right)>y\right\}$ for arbitrarily large $y$. This implies

[TABLE]

for arbitrarily large $y$. That means there exist arbitrarily large $y\in\left[0,\infty\right)$ and $x\in\left[0,\infty\right)$ fulfilling

[TABLE]

On the other hand we have that $f\left(x\right)=x^{\gamma\delta}L^{\gamma}\left(x^{\delta}\right)$ and

[TABLE]

This implies $f\left(\left(1+\epsilon\right)x\right)\geq\left(1+\epsilon/2\right)^{\gamma\delta}f\left(x\right)$, for $x$ sufficiently large. Since $f$ tends to infinity, this contradicts (41) and hence the assumption. ∎

We conclude this section with calculating the expectation of the truncated observable $\prescript{u_{n}}{}{\mathrlap{\smash{\chi}}\phantom{\chi}}$.

Lemma 3.18**.**

Let $\chi$ be such that $\mu(\chi>x)=L(x)/x^{\alpha}$ with $L$ a slowly varying function and $0<\alpha<1$. Further, let $\left(u_{n}\right)$ be a non-negative sequence with $\lim_{n\rightarrow\infty}u_{n}=\infty$. Then

[TABLE]

Proof.

First note that (43) follows immediately from (42). To see (42) note

[TABLE]

Also, $\left[x^{1-\alpha}L\left(x\right)\right]_{0}^{u_{n}}=u_{n}^{1-\alpha}L\left(u_{n}\right)$. To estimate the first summand of (44) we apply Karamata’s theorem, see for example [BGT87, Theorem 1.5.11], and obtain

[TABLE]

Hence,

[TABLE]

(43) follows immediately from (42) by noting that $\int\mathsf{T}_{n}^{u_{n}}\chi\mathrm{d}\mu=n\cdot\int\prescript{u_{n}}{}{\mathrlap{\smash{\chi}}\phantom{\chi}}\mathrm{d}\mu$. ∎

4. Proofs of main theorems

4.1. Proof of the exponential inequality

We will give here the proof of Lemma 2.9 in a series of lemmas. The proof applies the analytic perturbation theory discussed in Section 3.2, particularly Lemma 3.5 and Lemma 3.7. We start with a generalization of Doob’s inequality in order to estimate $\mu\left(\max_{i\leq n}\left|\mathsf{S}_{i}\varphi_{n}\right|\geq u_{n}\right)$ in Lemma 4.2.

Lemma 4.1** (Generalized inequality of Doob).**

Let $\left(M_{n}\right)_{n\in\mathbb{N}}$ be a martingale, then we have for every convex function $g:\mathbb{R}\to\mathbb{R}$ and every $x>0$ that

[TABLE]

Proof.

The proof of this theorem is the same as of the usual inequality of Doob (see [Kle07, Theorem 11.2] for example) using a general convex function instead of $g\left(x\right)=x^{p}$, i.e. by [Kle07, Theorem 9.35] we have that $\left(g\left(M_{n}\right)\right)_{n\in\mathbb{N}}$ is a submartingale if $\left(M_{n}\right)_{n\in\mathbb{N}}$ is a martingale, $g\colon\mathbb{R}\to\mathbb{R}$ is convex, and \mathbb{E}\,\big{(}g\left(M_{n}\right)^{+}\big{)}<\infty, for all $n\in\mathbb{N}$. We can assume the last inequality here. Otherwise the inequality in (45) holds trivially. Furthermore, [Kle07, Lemma 11.1] states that for a submartingale $\left(X_{n}\right)$ and $y>0$ we have by setting $X_{n}^{*}\coloneqq\sup_{k\leq n}\left|X_{k}\right|$ that $y\cdot\mathbb{P}\left(\left|X_{n}^{*}\right|\geq y\right)\leq\mathbb{E}\left(\left|X_{n}\right|\cdot\mathbbm{1}_{\left\{X_{n}^{*}\geq y\right\}}\right)$. Setting $y\coloneqq g\left(x\right)$ and $X_{n}\coloneqq g\left(\left|M_{n}\right|\right)$ yields the statement of the lemma. ∎

Lemma 4.2**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$. Then there exists $C>0$ such that for all $\varphi\in\mathcal{F}$ fulfilling $\int\varphi\mathrm{d}\mu=0$, all $n\in\mathbb{N}$, and all $t>0$

[TABLE]

Proof.

First define $\mathsf{R}\coloneqq\sum_{n=1}^{\infty}\widehat{T}^{n}$. We will now estimate $\mathsf{R}\varphi$ for $\int\varphi\mathrm{d}\mu=0$. We notice that for all $A\in\mathcal{A}$ we have

[TABLE]

and from Lemma 3.4 it follows that there exist $R>0$ and $0<\tau<1$ such that $\int\widehat{T}^{n}\varphi\cdot\mathbbm{1}_{A}\mathrm{d}\mu\leq R\cdot\tau^{n}\cdot\left\|\varphi\right\|\cdot\left|\mathbbm{1}_{A}\right|_{1}$. We set $C\coloneqq 4R\cdot\sum_{n=1}^{\infty}\tau^{n}$ and have $\int\mathsf{R}\varphi\mathbbm{1}_{A}\mathrm{d}\mu\leq C/4\cdot\left\|\varphi\right\|\cdot\left|\mathbbm{1}_{A}\right|_{1}<\infty$, for all $A\in\mathcal{A}$. Thus,

[TABLE]

Let $h\coloneqq\varphi+\mathsf{R}\varphi-\left(\mathsf{R}\varphi\right)\circ T$. From (47) and $\left|\cdot\right|_{2}\leq\left\|\cdot\right\|$ we have that $h\in\mathcal{L}^{2}$. By construction we have that $\mathsf{S}_{n}h=\sum_{j=0}^{n-1}h\circ T^{j}$ is a martingale. Further, $\varphi=h+\left(\mathsf{R}\varphi\right)\circ T-\mathsf{R}\varphi$ and thus,

[TABLE]

We have that

[TABLE]

Applying the generalized inequality of Doob, see Lemma 4.1, the fact that $\mathsf{S}_{n}h$ is a martingale, and (47) yields for all $t>0$

[TABLE]

Furthermore, we obtain by the definition of $h$ and (47) for the second factor that

[TABLE]

Combining this with (48) yields for all $t>0$

[TABLE]

Analogously, we obtain

[TABLE]

Thus, the statement of the lemma follows. ∎

Our next lemma estimates the second factor of (46).

Lemma 4.3**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$. Let $\lambda_{z}$ and $\varkappa^{\prime}$ be given as in Lemma 3.7. Then there exist $K^{\prime},N>0$ such that for all $\varphi\in\mathcal{F}$ fulfilling $\int\varphi\mathrm{d}\mu=0$, all $n\in\mathbb{N}_{>N}$, and all $t\in\left(0,\varkappa^{\prime}/\left\|\varphi\right\|\right)$

[TABLE]

Proof.

We will first show

[TABLE]

by induction. The base case is obvious. Hence, assume that (49) holds for $n-1$. This in combination with (1) yields for all $g\in\mathcal{L}^{1}$ that

[TABLE]

Since we have by Lemma 3.5 that $\widehat{T}_{t}^{n}=\lambda_{t}^{n}P_{t}+N_{t}^{n}$, it follows that

[TABLE]

and thus using (2) yields

[TABLE]

We observe by the separation of the spectrum given in Lemma 3.5 that $\left|\lambda_{t}\right|>1-\theta\geq 1-\left(1-\rho(N_{0})\right)/2\geq 1/2$. Combining this with (2) and (3) and the assumption $\left|t\right|\cdot\left\|\varphi\right\|<\varkappa^{\prime}$ yields

[TABLE]

Combining this calculation with (50) yields

[TABLE]

Applying Lemma 3.7 gives the statement of the lemma with $K^{\prime}=2\cdot K_{0}\cdot\exp\left(\varkappa^{\prime}\right)+1$. ∎

The next lemma gives an estimate of the $n$th derivative of $\lambda_{t}$ at $t=0$ which will help us later to estimate $\lambda_{t}$.

Lemma 4.4**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|\right)$ fulfill Property $\mathfrak{C}$. Let $\lambda_{z}$ be the eigenvalue of the perturbed transfer operator $\widehat{T}_{z}$ given in Lemma 3.5. Then we have $\lambda_{0}^{\left(0\right)}=1,\lambda_{0}^{\left(1\right)}=0$, and for $n\geq 2$ there exists $\eta<\infty$ such that for all $\varphi\in\mathcal{F}$ with $\int\varphi\mathrm{d}\mu=0$ we have that

[TABLE]

Proof.

We assume in the sequel that $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa$ with $\varkappa$ as in Lemma 3.5. By Lemma 3.5 we have that $\widehat{T}_{z}P_{z}=\left(\lambda_{z}P_{z}+N_{z}\right)P_{z}=\lambda_{z}P_{z}$ with all parts being analytic. Differentiating both sides $n$ times and using (30) we get by induction that

[TABLE]

Applying (53) to the constant one function, multiplying by a test function $h$, and integrating yields

[TABLE]

Recalling the connection between $\widehat{T}$ and $T$ from (1) yields

[TABLE]

Setting $h\equiv 1$ we have that

[TABLE]

If we set $n=1$, then

[TABLE]

and thus, with $P_{0}h=\int h\mathrm{d}\mu$, see Lemma 3.3, i.e. $P_{0}1=1$, $\int\varphi\mathrm{d}\mu=0$, and $\lambda_{0}=1$ it follows that $\lambda_{0}^{\left(1\right)}=0$.

From (54) it follows then for $n\geq 2$

[TABLE]

In the next steps we will estimate $P_{0}^{\left(k\right)}1\left(\omega\right)$. Remember that by (51) we have $\left|P_{z}1\right|_{\infty}\leq K^{\prime}$ if $\left|z\right|\cdot\left\|\varphi\right\|<\varkappa$. Since $P_{z}$ is analytic on a circle around zero with radius $r\leq\varkappa/\left\|\varphi\right\|$ it follows by Cauchy’s integral formula and the maximum principle that

[TABLE]

We notice that

[TABLE]

Hence, we can estimate the first sum of (55) by

[TABLE]

A further estimate yields

[TABLE]

since $a\geq 2$.

In the sequel we will prove that (52) holds for $\eta=3a$ using an induction argument. The base case for $n=2$ is obvious since the minuend in (55) equals zero in that case and thus $\lambda^{(2)}\leq 2!\cdot a^{2}\cdot\left|\varphi\right|_{1}\cdot\left\|\varphi\right\|\leq 2!\cdot b^{2}\cdot\left|\varphi\right|_{1}\cdot\left\|\varphi\right\|$. By assuming that (52) holds for all $k\leq n-1$ we have for the second sum in (55) that

[TABLE]

Furthermore,

[TABLE]

Combining this with (55), (56), (57), and (58) yields

[TABLE]

which gives the statement of the lemma. ∎

Proof of Lemma 2.9.

We assume that $t\in\left(0,\varkappa^{\prime}/\left\|\varphi\right\|\right)$. Using a combination of Lemma 4.2 and Lemma 4.3 implies that there exists $N>0$ as in Lemma 4.3 such that for all $n\in\mathbb{N}_{>N}$

[TABLE]

with $C$ as in Lemma 4.2 and $K^{\prime}$ as in Lemma 4.3. Furthermore, by our choice of $t$ we have that $C\cdot\left\|\varphi\right\|\cdot t$ is bounded by $C\cdot\varkappa^{\prime}$. This implies

[TABLE]

with $K\coloneqq 2K^{\prime}\cdot\exp\left(C\cdot\varkappa^{\prime}\right)$.

From Lemma 3.5 and Lemma 4.4 it follows for $t\in\left(0,\varkappa^{\prime}/\left\|\varphi\right\|\right)$ that

[TABLE]

For the following we assume that $t\cdot\left\|\varphi\right\|\leq\min\left\{\varkappa^{\prime},\left(2\eta\right)^{-1}\right\}$. This implies $\eta\cdot\left\|\varphi\right\|\cdot t<1/2$ and we can apply the geometric series formula obtaining

[TABLE]

Thus,

[TABLE]

Combining (59) and (60) yields

[TABLE]

We set for the following

[TABLE]

and

[TABLE]

This definition of $t$ ensures that $\eta\cdot\left\|\varphi\right\|\cdot t<1/2$ and $t\cdot\left\|\varphi\right\|\leq\varkappa^{\prime}$. Assume the minimum is attained at the first entry in (61). Then

[TABLE]

If the minimum in (61) is attained at the second entry, then $t\cdot u>1/2\cdot t^{2}\cdot n\cdot 2\eta^{2}\cdot\left|\varphi\right|_{1}\cdot\left\|\varphi\right\|$ and thus

[TABLE]

These two considerations give the statement of the lemma. ∎

4.2. Proof of theorems concerning the truncated sum $\mathsf{T}_{n}^{f_{n}}\chi$

In this section we will prove the theorems given in Section 2.2. We will start with a lemma to be used in their proofs.

Lemma 4.5**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$. Then there exist constants $N^{\prime}\in\mathbb{N}$ and $E,K>0$ such that for all $\epsilon\in(0,E)$, $r>0$ with $F(r)>0$, and $n\in\mathbb{N}_{>N^{\prime}}$

[TABLE]

Proof.

We denote for the following $\prescript{r}{}{\mathrlap{\smash{\overline{\chi}}}\phantom{\overline{\chi}}}\coloneqq\prescript{r}{}{\mathrlap{\smash{\chi}}\phantom{\chi}}-\int\prescript{r}{}{\mathrlap{\smash{\chi}}\phantom{\chi}}\mathrm{d}\mu$. Applying Lemma 2.9 yields

[TABLE]

for $n\geq N$ given in Lemma 2.9. We have that $n\cdot\left|\prescript{r}{}{\mathrlap{\smash{\overline{\chi}}}\phantom{\overline{\chi}}}\right|_{1}\geq 1/2\cdot\int\mathsf{T}^{r}_{n}\chi\mathrm{d}\mu\geq\epsilon\int\mathsf{T}^{r}_{n}\chi\mathrm{d}\mu$ if $\epsilon\leq 1/2$. Furthermore, from (4) we obtain

[TABLE]

Thus,

[TABLE]

for $n\geq N$ given in Lemma 2.9.

If $\epsilon$ is sufficiently small, then $\epsilon^{2}\cdot U_{1}\cdot\left(2\cdot\left(K_{1}+\left\|1\right\|\right)\cdot r\right)^{-1}$ and we obtain the statement of the lemma. ∎

Proof of Theorem 2.4.

We assume that (16) is fulfilled for a sequence $\left(f_{n}\right)$ with $F\left(f_{n}\right)>0$, for all $n\in\mathbb{N}$. Further, we note that this condition implies that for all $\epsilon>0$ there exists $N\in\mathbb{N}$ such that

[TABLE]

for some $\psi\in\Psi$ and all $n\geq N$. Fix $\epsilon>0$ and apply Lemma 4.5 to obtain

[TABLE]

for $n$ sufficiently large. From (62) we can conclude that the right hand side is summable and obtain by the Borel-Cantelli lemma that $\mu\left(\left|\mathsf{T}_{n}^{f_{n}}\chi-\int\mathsf{T}_{n}^{f_{n}}\chi\mathrm{d}\mu\right|\geq\epsilon\int\mathsf{T}_{n}^{f_{n}}\chi\mathrm{d}\mu\text{ i.o.}\right)=0.$ Since $\epsilon>0$ is arbitrary, it follows that $\left|\mathsf{T}_{n}^{f_{n}}\chi-\int\mathsf{T}_{n}^{f_{n}}\chi\mathrm{d}\mu\right|=o\left(\int\mathsf{T}_{n}^{f_{n}}\chi\mathrm{d}\mu\right)$ almost surely and hence the assertion of the theorem. ∎

Proof of Theorem 2.5.

We define the sequences $\left(g_{n}\right)_{n\in\mathbb{N}}$ and $\left(\overline{g}_{n}\right)_{n\in\mathbb{N}}$ with $g_{n}\coloneqq\max\left\{f_{n},n\right\}$ and $\overline{g}_{n}\coloneqq\min\left\{f_{n},n\right\}$. Further, set $I_{j}\coloneqq\left[2^{j},2^{j+1}-1\right]$ for $j\in\mathbb{N}$ and for given $\epsilon>0$ we set $\rho_{k}\coloneqq\rho_{k}\left(\epsilon\right)\coloneqq\left(1+\epsilon\right)^{k}$ and the sequences $\left(q_{j}\right)_{j\in\mathbb{N}}$ and $\left(r_{j}\right)_{j\in\mathbb{N}}$ as

[TABLE]

These numbers are chosen such that

[TABLE]

We will split the proof of this theorem as follows:

1. (A)

There exist $E^{\prime},N>0$ such that for all $j\geq N$ and $\epsilon<E^{\prime}$

[TABLE] 2. (B)

We have that

[TABLE] 3. (C)

For all $\epsilon>0$ we have

[TABLE]

Applying the Borel-Cantelli lemma on (B) and combining this with (A) using (63) yields

[TABLE]

Combining this with (C) gives the statement of the theorem.

Proof of (A). We have that (42) implies

[TABLE]

for $k$ tending to infinity. Choosing $k$ such that $\rho_{k}\leq g_{n}\leq\rho_{k+1}$ implies

[TABLE]

for $n$ sufficiently large since $g_{n}$ tends to infinity. We assume for the following that $\epsilon<1/4$. Then we obtain the following implications

[TABLE]

and thus

[TABLE]

Analogously to the situation above we obtain

[TABLE]

and thus

[TABLE]

Furthermore, we have for every $m\in I_{j}$ and $k\in\mathbb{N}$ that

[TABLE]

Combining (65), (66), and (67) yields (A).

Proof of (B). We have that $\prescript{\rho_{k}}{}{\mathrlap{\smash{\chi}}\phantom{\chi}}\leq\rho_{k}$. Hence, we can apply Lemma 4.5 assuming that $\epsilon<E$ to the sum $\mathsf{T}_{n}^{\rho_{k}}\chi$ and obtain for $j$ sufficiently large

[TABLE]

Using (43) we have for $k$ tending to infinity that

[TABLE]

Hence, we obtain by (68) that

[TABLE]

We estimate

[TABLE]

for $j$ sufficiently large. We obtain by factoring out that

[TABLE]

For $k$ sufficiently large we have ${L((1+\epsilon)^{k})}/{L((1+\epsilon)^{k+1})}>{(1+\epsilon)^{-\alpha/2}}$ and hence we have for $k>q_{j}$ and $j$ sufficiently large ${L((1+\epsilon)^{k})}/{L((1+\epsilon)^{r_{j}})}>{(1+\epsilon)^{-\alpha\cdot(r_{j}-k)/2}}.$ Applying this to (70) and defining $\epsilon_{1}\coloneqq(1+\epsilon)^{\alpha/2}-1$ yields

[TABLE]

for $k>q_{j}$ and $j$ sufficiently large. Hence, we can understand the second factor of (69) as a geometric series and obtain

[TABLE]

Furthermore, we have that $\left(1+\epsilon\right)^{r_{j}-1}<\max_{n\in I_{j}}g_{n}\leq\left(1+\epsilon\right)^{r_{j}}$. Since $F$ with $F\left(x\right)=1-L\left(x\right)/x^{\alpha}$ is a distribution function, $L\left(x\right)/x^{\alpha}$ is decreasing. Thus, the slow variation of $L$ implies

[TABLE]

for $j$ sufficiently large. To continue we state the following lemma, which is [KS17, Lemma 5].

Lemma 4.6**.**

Let $a,b>1$ and $\psi\in\Psi$. Then there exists $\omega\in\Psi$ such that

[TABLE]

From the above observations and Lemma 4.6 it follows that there exists $\widetilde{\kappa}>0$ and $\widetilde{\psi}\in\Psi$ such that

[TABLE]

for all $m\in I_{j}$. Thus, the existence of $\psi\in\Psi$ such that (17) holds implies together with (72)

[TABLE]

In particular we have

[TABLE]

and thus we have from (71) that

[TABLE]

for $j$ sufficiently large.

On the other hand we have that

[TABLE]

for $j$ sufficiently large. Hence, we have for the first factor of (69) that

[TABLE]

for $r_{j}$ sufficiently large. Inserting (73) and (74) in (69) yields

[TABLE]

for $j$ and thus $q_{j}$ sufficiently large. Next we show that the above expression is summable in $j$. (17) implies that there exists $\psi\in\Psi$ such that

[TABLE]

for all $n\in\mathbb{N}$. Applying Lemma 4.6 with $a=\mathrm{e}$ and $b=2$ yields that this implies the existence of $\omega\in\Psi$ such that

[TABLE]

This implies

[TABLE]

and thus,

[TABLE]

Since $\omega\in\Psi$, this is equivalent to

[TABLE]

Thus,

[TABLE]

i.e. the statement of (B).

Proof of (C). We note that (42) implies that there exists $R\in\mathbb{\mathbb{\mathbb{\mathbb{R}}}}_{>0}$ such that

[TABLE]

for all $r\geq R$. With

[TABLE]

we have that

[TABLE]

and since we have for $\psi\left(n\right)=n^{2}$ that

[TABLE]

by Theorem 2.4 we find that the statement of (C) holds. ∎

4.3. Proof of the large deviation result

In this section we will prove Lemma 2.8. The proof is similar to the proof of [KS17, Lemma 11]. However, due to dependence some additional argumentations are required. In order to increase the readability, we will formulate this proof to be self contained.

Proof of Lemma 2.8.

Let $I_{n}\coloneqq\left[2^{n},2^{n+1}-1\right]$ and

[TABLE]

for $\psi\in\Psi$ and let $\nu:\mathbb{N}\to\mathbb{N}$ be defined as $\nu\left(n\right)\coloneqq\left\lfloor\log_{2}n\right\rfloor$. Further let

[TABLE]

and

[TABLE]

Furthermore, define for $l,n\in\mathbb{N}$ fulfilling $\Delta_{n,l}\neq\varnothing$ the truncated observables

[TABLE]

The definition of those observables ensures

[TABLE]

We will show (20) by separately showing the following:

1. (A)

For all $i\in I_{n}$ and $V>0$ we have that

[TABLE]

and

[TABLE] 2. (B)

There exists $V>0$ such that

[TABLE]

and

[TABLE] 3. (C)

[TABLE]

Applying the Borel-Cantelli lemma on (B) and combining this with (A) yields

[TABLE]

This together with (C) yields the statement of the lemma.

Proof of (A). Let us assume $i\in I_{n}$ and

[TABLE]

This implies by the construction of $f_{n,l}$ and $g_{n,l}$ that

[TABLE]

We can conclude from (76) that $\left(p_{i}-\int f_{n,l}\mathrm{d}\mu\right)\cdot i\leq{i}/{2^{n+1}}\leq 1$ and furthermore $l\leq p_{i}\cdot 2^{n+1}\leq 2\cdot p_{i}\cdot i$. Thus,

[TABLE]

if $V\geq 2$. On the other hand by (76) we can conclude that $\left(\int g_{n,l}\mathrm{d}\mu-p_{i}\right)\cdot i\leq i/2^{n+1}\leq 1$ and $l\leq p_{i}\cdot 2^{n+1}\leq 2\cdot p_{i}\cdot i$. Hence,

[TABLE]

if $V\geq 2$. Combining (77) and (78) gives the statement of (A).

Proof of (B). To ease notation we define

[TABLE]

We aim to apply Lemma 2.9 and note that by (76)

[TABLE]

if $V>1$. Using Lemma 2.9 this implies

[TABLE]

for $n$ sufficiently large. Furthermore, we have by (5)

[TABLE]

Hence,

[TABLE]

If we set

[TABLE]

then

[TABLE]

We estimate

[TABLE]

This estimation holds for $k$ sufficiently large. Furthermore,

[TABLE]

for $n$ and thus $\kappa_{n}$ sufficiently large. Further, by construction

[TABLE]

Furthermore, we can conclude from Lemma 4.6 that there exists $\omega\in\Psi$ such that

[TABLE]

Hence, combining this with (82), (83), and (84) yields

[TABLE]

(75) follows from analogous calculations.

Proof of (C). For $n\in\mathbb{N}$ define the random variables

[TABLE]

which imply $\int h_{n}\mathrm{d}\mu\leq\kappa_{n}/2^{n}$. To ease notation set $\overline{h}_{n}\coloneqq h_{n}-\int h_{n}\mathrm{d}\mu$. We have for $i\in I_{n}$ and $p_{i}\leq\kappa_{n}/2^{n}$ that $\int h_{n}\mathrm{d}\mu\geq p_{i}$ and thus

[TABLE]

implies

[TABLE]

which yields

[TABLE]

if $V\geq 4$. On the other hand we have for $i\in I_{n}$ and $p_{i}\leq\kappa_{n}/2^{n}$ that

[TABLE]

if $V\geq 1$. Combining this with the above considerations yields

[TABLE]

Likewise as in the calculations leading to (81) replacing $l$ by $\kappa_{n}$ in (79) and (80), it follows for sufficiently large $n$ that

[TABLE]

Applying (84) and (85) yields

[TABLE]

The Borel-Cantelli lemma implies the statement or (C). ∎

4.4. Proofs of theorems concerning the trimmed sum $\mathsf{S}_{n}^{b_{n}}\chi$

In this section we will give the proof of Theorem 1.3, its Corollary 1.5 and Theorem 1.7. The main idea of all proofs is to use statements which state that under the given assumptions Properties $\bm{A}$ and $\bm{B}$ are fulfilled and consequently Lemma 2.3 can be applied.

Proof of Theorem 1.3.

Our strategy is to show that all properties are fulfilled such that we can apply Lemma 2.3.

If (6) holds, then also (16) for the choice $\psi(n)=n^{2}$ which fulfills $\psi\in\Psi$. Thus, Theorem 2.4 implies that Condition $\bm{A}$ holds.

Furthermore, the definition of $b_{n}$, the above choice of $\psi$ combined with Lemma 2.8, and setting $W=2V$ yield that Condition $\bm{B}$ holds.

Finally, by the definition of $b_{n}$ we have that $\gamma_{n}<W\cdot\max\{a_{n}^{1/2+\epsilon}\cdot\left(\log\log n\right)^{1/2-\epsilon},\log\log n\}+1$. Hence, (6) implies that (13) holds. ∎

Proof of Corollary 1.5.

The proof of Theorem 1.5 is basically the same as the proof of [KS17, Theorem A] applying Theorem 1.3 instead of [KS17, Theorem B]. ∎

Proof of Theorem 1.7.

In the first part of the proof we will show (8). We define

[TABLE]

with $c$ as in (19). We have that $F(F^{\leftarrow}(x)-1)\leq x$. Since $c$ is monotonically increasing in its first variable, it follows that

[TABLE]

Hence, Lemma 2.8 implies that the pair $((f_{n}),(\gamma_{n}))$ fulfills Property $\bm{B}$.

In the following we will show that this choice of $(f_{n})$ fulfills Property $\bm{A}$. Assume that $\left(b_{n}\right)$ fulfills (7) for some $\psi\in\Psi$. If we consider $c_{\epsilon,\psi}\left(k,n\right)$ in (19) with the same $\psi$, then $c\left(b_{n},n\right)=b_{n}^{1/2+\epsilon}\log\psi\left(\left\lfloor\log n\right\rfloor\right)^{1/2-\epsilon}$ and thus $W\cdot c\left(b_{n},n\right)=o\left(b_{n}\right)$.

In the next steps we aim to prove that $(1-F\left(f_{n}\right))\cdot n\sim b_{n}$. We have on the one hand that $F(F^{\leftarrow}(x))\geq x$ and on the other hand $F(F^{\leftarrow}(x)-1)\leq x$. This yields

[TABLE]

The second asymptotic holds because by assumption $b_{n}=o(n)$ which implies that $(f_{n})$ tends to infinity.

The above observation combined with (7) implies

[TABLE]

which is equivalent to (17) and Theorem 2.5 states that under this condition Property $\bm{A}$ holds.

In the last steps we will prove (13). By (43) we have that

[TABLE]

Since $b_{n}\sim n\cdot(1-F(f_{n}))$, it follows that $\gamma_{n}=o\left(n\cdot\left(1-F\left(f_{n}\right)\right)\right)$. This implies (13). Hence, we can apply Lemma 2.3 and obtain the first part of the theorem.

In the next steps we will show the asymptotic given in (9) by finding an asymptotic equivalent sequence in terms of $\left(b_{n}\right)$ for $\int\mathsf{T}_{n}^{f_{n}}\chi\mathrm{d}\mu$. We have by (43) and the definition of $F$ and $(f_{n})$ that

[TABLE]

Set for the following $u_{n}\coloneqq n/\left(b_{n}-W\cdot c\left(b_{n},n\right)\right)$ and the function $G:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ with $G\left(x\right)\coloneqq 1/\left(1-F\left(x\right)\right)=x^{\alpha}/L\left(x\right)$. Then

[TABLE]

Since $u_{n}$ tends to infinity, we can apply Lemma 3.17 on $G$ and obtain

[TABLE]

This and the fact that $b_{n}\sim b_{n}-W\cdot c\left(b_{n},n\right)$ yields

[TABLE]

Hence, applying (86) and (88) to (87) yields (9). ∎

4.5. Proof of main example

Proof of Propositions 1.11 and 1.12.

We have for all $f,g\in BV$ that

[TABLE]

and thus, it can be deduced that $BV$ is a Banach algebra which contains the constant functions.

Further define $h:\Omega\to\mathbb{R}_{\geq 0}$ as a function fulfilling $\left|h\right|_{\infty}\leq 1$, $\mathsf{V}\left(h\right)<\infty$, and $h\lvert_{\Omega\backslash\Omega^{\prime}}=0$. With this function we define an operator $\mathcal{P}:\mathcal{L}^{1}\to\mathcal{L}^{1}$ by

[TABLE]

Furthermore, let $h$ be such that the adjoint operator $\mathcal{P}^{*}$ preserves the Lebesgue measure $\lambda$, i.e. $\mathcal{P}^{*}\left(\lambda\right)\left(f\right)=\lambda\left(\mathcal{P}f\right)=\lambda\left(f\right)$ for all $f\in\mathcal{L}^{1}$.

Now, given all the properties of $T$, [Ryc83, Theorem 1] states that the operator $\mathcal{P}$ fulfills all the properties for a spectral gap given in Definition 3.1 except that there might be finitely many mutually orthogonal one dimensional projections. Combining this with [Ryc83, Remark 4b] gives that in case the system is topologically mixing there is only one projection corresponding to the eigenvalue $1$. Furthermore, [Ryc83, Theorem 3] implies in the topologically mixing case the existence of a function $f$ such that $\mathcal{P}f=f$ and thus the existence of a measure $\mu$ absolutely continuous with respect to $\lambda$.

Finally, (10) and (11) ensure that (4) and (5) are fulfilled with $K_{1}\coloneqq 1+\widetilde{K}_{1}$ and $K_{2}\coloneqq 1+\widetilde{K}_{2}$. ∎

Acknowledgements

This research was supported by the German Research Foundation (DFG) grant Renewal Theory and Statistics of Rare Events in Infinite Ergodic Theory (Geschäftszeichen KE 1440/2-1). TS was partly supported by the Studienstiftung des Deutschen Volkes.

We thank Gerhard Keller for referring us to the paper of Rychlik.

Acknowledgements

We thank Gerhard Keller for referring us to the paper of Rychlik.