Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type

TL;DR

This paper establishes strong laws of large numbers with intermediate trimming for Birkhoff sums on subshifts of finite type, extending previous results and introducing new function spaces.

Contribution

It introduces intermediate trimming laws for Birkhoff sums on subshifts of finite type and develops the space of quasi-Hölder functions for these systems.

Findings

Proves strong laws of large numbers with intermediate trimming.

Provides examples of St. Petersburg type distributions for Markov measures.

Extends trimming results from interval maps to subshifts of finite type.

Abstract

We prove strong laws of large numbers under intermediate trimming for Birkhoff sums over subshifts of finite type. This gives another application of a previous trimming result only proven for interval maps. In case of Markov measures we give a further example of St.\ Petersburg type distribution functions. To prove these statements we introduce the space of quasi-H\"older continuous functions for subshifts of finite type.

Equations

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

$$X:=\left\{x={\left(x_n\right)}_{n\in {\mathbb{N}}_0}:x_n\in \mathbb{A},A\left(x_n,x_{n+1}\right)=1\right\},$$

$$\mathrm{osc}\left(h,C\right):=\underset{x\in C}{\mathrm{esssup}}h\left(x\right)-\underset{x\in C}{\mathrm{essinf}}h\left(x\right)$$

$$d_2(x,y):=\inf \{\mu (C):x,y\in C,C\subset X\text{mbox}cylinderset\}$$

$${\left|h\right|}_{{ϵ}_0}:=\underset{0<ϵ\le {ϵ}_0}{\sup }\frac{\int \mathrm{osc}\left(h,B\left(ϵ,x\right)\right)\mathrm{d}\mu }{ϵ}$$

$${\Vert h\Vert }_{{ϵ}_0}:={\left|h\right|}_{\infty }+{\left|h\right|}_{{ϵ}_0}.$$

$$\mathsf{H}:=\left\{h\in {\mathcal{L}}^{\infty }\left(X,\mathbb{R}\right):{\left|h\right|}_{{ϵ}_0}<\infty \right\}.$$

$${\left|\text{prescript}\ell \chi \right|}_{{ϵ}_0}\le K_1\ell .$$

$${\left|\text{mathbbm}{1}_{\{\chi \ge \ell \}}\right|}_{{ϵ}_0^{\prime }}\le K_2.$$

$${\left|f\right|}_{\theta }:=\underset{x,y\in X}{\sup }\frac{\left|f\left(x\right)-f\left(y\right)\right|}{d_1\left(x,y\right)}$$

$$\underset{x\to \infty }{\lim }L\left(x\right)\cdot L^{\#}\left(xL\left(x\right)\right)=1=\underset{x\to \infty }{\lim }{L}^{\#}\left(x\right)\cdot L\left(x{L}^{\#}\left(x\right)\right).$$

$$\Psi :=\left\{u:\mathbb{N}\to {\mathbb{R}}_{>0}:\sum _{n=1}^{\infty }\frac{1}{u\left(n\right)}<\infty \right\}.$$

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

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

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

$$\chi \left(\left(x_n\right)\right):={\eta }^{\min \left\{j\in \mathbb{N}:x_j\ne {\mathfrak{a}}_1\right\}-1}.$$

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

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

$$d_n\sim \frac{\eta }{q\cdot \eta -1}\cdot R^{-\log \eta /\log q}\cdot {\left(1-q\right)}^{1+\log \eta /\log q}\cdot n^{-\log \eta /\log q}\cdot b_n^{1+\log \eta /\log q}.$$

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

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

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

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

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

$$\Vert \text{mathbbm}{1}_{\{\chi =\ell \}}\Vert \le K_3.$$

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

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

$$d_n=\frac{\eta }{q\cdot \eta -1}\cdot {\left(R\cdot q\right)}^{-\log \eta /\log q}\cdot {\left(1-q\right)}^{1+\log \eta /\log q}\cdot n^{-\log \eta /\log q}\cdot b_n^{1+\log \eta /\log q}.$$

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

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

Full text

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).

1. Introduction and statement of main results

If we consider an ergodic dynamical system $\left(\Omega,\mathcal{A},T,\mu\right)$ with $\mu$ a probability measure and a stochastic process given by the Birkhoff sum $\mathsf{S}_{n}\chi\coloneqq\sum_{k=1}^{n}\chi\circ T^{k-1}$ with $\mathsf{S}_{0}\chi=0$ for some measurable function $\chi:\Omega\to\mathbb{R}_{\geq 0}$, then with respect to a 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.) $\lim_{n\to\infty}\mathsf{S}_{n}\chi/n=\int\chi\mathrm{d}\mu$, i.e.  the strong law of large numbers is fulfilled, whereas for the case that $\chi$ is non-integrable, Aaronson ruled out the possibility of a strong law of large numbers, see [Aar77].

However, in certain cases it is possible to obtain 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 $\pi\in\mathcal{S}_{n}$ of $\left\{0,\ldots,n-1\right\}$ with $\chi\circ T^{\pi\left(1\right)}\geq\chi\circ T^{\pi\left(2\right)}\geq\ldots\geq\chi\circ T^{\pi\left(n\right)}$ and for given $b\in\mathbb{N}_{0}$ we define

[TABLE]

In [KS18] the authors considered a general setting of dynamical systems $(\Omega,\mathcal{A},\mu,T)$ obeying a set of conditions given in Property $\mathfrak{D}$, see Definition 2.3. One key assumption is that the transfer operator (see (5)) fulfills a spectral gap property on $\mathcal{F}$, a subset of the measurable functions forming a Banach algebra with respect to a norm $\left\|\cdot\right\|$. For such systems the authors proved intermediately trimmed strong laws, i.e. the existence of a sequence of natural numbers $(b_{n})$ tending to infinity with $b_{n}=o(n)$ and a norming sequence $(d_{n})$ such that $\lim_{n\to\infty}\mathsf{S}_{n}^{b_{n}}\chi/d_{n}=1$ a.s. One interesting case is the example of regularly varying tail distributions for which the same trimming sequence can be chosen as in the i.i.d. case. The key example studied in [KS18] are piecewise expanding interval maps.

It is the aim of the present paper to adapt these results to subshifts of finite type. It turns out that in contrary to the example of piecewise expanding interval maps it is not immediately clear how to apply the results from [KS18] to subshifts of finite type as the usually considered space of Lipschitz continuous functions does not fulfill all required properties of Property $\mathfrak{D}$. In particular, Property $\mathfrak{D}$ requires $\left\|\mathbbm{1}_{\left\{\chi>\ell\right\}}\right\|\leq K$ uniformly in $\ell$ which can not be fulfilled for an unbounded potential $\chi$ with respect to the Lipschitz norm, see Remark 1.2.

Instead we consider the Banach space of quasi-Hölder continuous functions which is larger than the space of Lipschitz continuous functions but still obeys a spectral gap property. A similar Banach space was first considered by Blank, see [Bla97, Chapter 2.3], and Saussol, see [Sau00], in the context of multidimensional expanding maps.

Furthermore, we prove some new limit theorems not given in [KS18] which we consider as particularly interesting for the application to observables on a subshift with a Gibbs-Markov measure, see Section 1.3.

As a side result we obtain a limit theorem for sums of the truncated random variables. Namely, for an observable $\chi:\Omega\to\mathbb{R}_{\geq 0}$ and a real valued sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$ we consider the truncated sum $\mathsf{T}_{n}^{f_{n}}\chi\coloneqq\sum_{k=1}^{n}\left(\chi\cdot\mathbbm{1}_{\{\chi\leq f_{n}\}}\right)\circ T^{k-1}$. These results are not only valid for the considered setting of subshifts of finite type but also for other dynamical systems fulfilling a spectral gap property and for i.i.d. random variables. This also improves a limit theorem for St. Petersburg games given in [GK11] and [Nak15] for i.i.d. random variables with stronger conditions on the truncation sequence, see Theorem 2.9 and Remark 2.10.

It is worth mentioning that proving limit theorems under trimming for non-integrable (independent and dependent) random variables has a long tradition and particularly for the i.i.d. case there is a vast literature. Here we will only mention results stating a generalized strong law of large numbers.

The first considered limit laws were lightly trimmed strong laws, i.e. the existence of $b\in\mathbb{N}$ independent of $n$ and a norming sequence $(d_{n})$ such that $\lim_{n\to\infty}\mathsf{S}_{n}^{b}/d_{n}=1$ holds almost surely.

Mori developed general conditions for lightly trimmed strong laws for i.i.d. random variables, see [Mor76], [Mor77]. These results have been generalized by Kesten and Maller, see [Mal84], and [KM92].

It became obvious by a result by Kesten, see [KM95], that light trimming is not always enough to prove strong laws of large numbers, as in particular weak laws of large numbers for i.i.d. random variables do not change under light trimming. One example for which no weak law of large numbers hold are regularly varying tail distributions with exponent $\alpha\in(-1,0)$. For such distribution functions an intermediately trimmed strong law can be easily deduced from results by Haeusler and Mason, see [HM87], and a lower bound for the trimming sequence $(b_{n})$ can be derived from a result by Haeusler, see [Hae93]. In [HM91] and [KS17] intermediately trimmed strong laws for other distribution functions are given.

The history for trimmed strong laws in the dynamical systems setting follows a similar line. One of the first investigated examples for the case of dynamical systems is the unique continued fraction expansion. Diamond and Vaaler showed in [DV86] a lightly trimmed strong law for the digits of the continued fraction expansion. Aaronson and Nakada extended the afore mentioned results by Mori to $\bm{\psi}$-mixing random variables in [AN03], i.e. they gave sufficient conditions for a lightly trimmed strong law to hold. In [Hay14] Haynes gave a quantitative strong law of large numbers under light trimming for certain classes of dynamical systems. The results in [AN03] and [Hay14] are also applicable for subshifts of finite type obeying some additional conditions.

However, for certain types of dynamical systems light trimming is not enough even though a lightly trimmed strong law would hold in the i.i.d. case for random variables with the same distribution function, see [AN03] and [Hay14]; see also [Sch18] for the corresponding intermediately trimmed strong law for such a system.

In contrast to the lightly trimmed case an intermediately trimmed strong law for regularly varying tail distributions with exponent $\alpha\in(-1,0)$ holds with the same trimming sequence $(b_{n})$ and norming sequence $(d_{n})$ for both i.i.d. random variables and a large class of dynamical systems, see [KS18]. As noted above we will prove that a large class of observables on subshifts of finite type can also fulfill these properties and the same intermediately trimmed strong law holds.

1.1. The space of quasi-Hölder continuous functions

In order to state our main theorem we first introduce the space of quasi-Hölder continuous functions on the subshift of finite type

[TABLE]

where $\mathbb{A}$ denotes a finite alphabet and $A\in\left\{0,1\right\}^{\mathbb{A}\times\mathbb{A}}$ is an irreducible and aperiodic matrix (see Definition 3.1). For the following we denote by $\mathbb{A}_{n}$ the set of all admissible sequences of length $n$ and for $\mathsf{A}\in\mathbb{A}_{n}$ we let $\left[\mathsf{A}\right]\subset X$ denote the cylinder set determined by $\mathsf{A}$ and let $X$ represent the cylinder set of the empty word. The $\sigma$-algebra generated by the set of cylinder sets will be denoted by $\mathcal{B}$. With $\sigma:X\to X$ we denote the left shift and we define a metric $d_{1}$ on $X$ by $d_{1}\left(x,y\right)\coloneqq\theta^{k}$, for some $\theta\in\left(0,1\right)$ where $k\in\mathbb{N}_{0}$ is the largest integer such that $x_{i}=y_{i}$ for all $i\leq k$. Next we fix a probability measure $\mu$ that is a $g$-measure such that the corresponding $g$-function (see Definition 3.4) is given by $\exp(f)$ for $f\colon X\to\mathbb{R}_{>0}$ being Lipschitz continuous with respect to the metric $d_{1}$. Note that a $g$-measure is always $\sigma$-invariant, atomless and assigns positive measure to every non-empty cylinder set.

For a measurable function $h\colon X\to\mathbb{R}_{\geq 0}$ the oscillation on $C\subset X$ is given by

[TABLE]

and we set $\operatorname{osc}\left(h,\varnothing\right)\coloneqq 0$. For the fixed measure $\mu$ we define the metric $d_{2}$ on $X$ given by

[TABLE]

and we let $B\left(\epsilon,x\right)$ denote the $\epsilon$-ball around $x$ with respect to this metric. Then for fixed $\epsilon_{0}\in\left(0,1\right)$ we define

[TABLE]

to obtain the norm

[TABLE]

We will show in Lemma 3.6 that $\left\|\cdot\right\|_{\epsilon_{0}}$ is indeed a norm. With this at hand we can define the space of quasi-Hölder continuous functions

[TABLE]

We remark here that the norm $\|\cdot\|_{\epsilon_{0}}$ depends on $\epsilon_{0}$, but $\mathsf{H}$ is independent of the choice of $\epsilon_{0}$.

Given these definitions we are able to state the setting for our main theorem.

Definition 1.1**.**

For a $\mathcal{B}$-measurable function $\chi\colon X\to\mathbb{R}_{\geq 0}$ we set $\prescript{\ell}{}{\mathrlap{\smash{\chi}}\phantom{\chi}}\coloneqq\chi\cdot\mathbbm{1}_{\left\{\chi\leq\ell\right\}}$ and say that $\left(X,\mathcal{B},\mu,\chi\right)$ fulfills Property $\mathfrak{F}$ if the following conditions hold:

• •

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

[TABLE]

• •

There exists $K_{2}>0$ and $\epsilon_{0}^{\prime}>0$ such that for all $\ell>0$

[TABLE]

Remark 1.2*.*

In [KS18] we impose Property $\mathfrak{D}$ (see Definition 2.3) as a sufficient condition for a trimmed strong law to hold, which in particular requires that there exists $K_{2}>0$ such that for all $\ell>0$ we have $\left\|\mathbbm{1}_{\left\{\chi>\ell\right\}}\right\|\leq K_{2}$, see (7). Now, the space of Lipschitz continuous functions (with respect to the metric $d_{1}$) is given by $F_{\theta}\left(X\right)\coloneqq\left\{f:X\to\mathbb{R}_{\geq 0}\colon\left|f\right|_{\theta}<\infty\right\}$, where

[TABLE]

and is equipped with the norm $\left\|f\right\|_{\theta}\coloneqq\left|f\right|_{\infty}+\left|f\right|_{\theta}$. From this it becomes apparent that we can not find an unbounded observable $\chi$ such that Property $\mathfrak{D}$ is fulfilled for $\left(X,\mathcal{B},\mu,\sigma,F_{\theta},\left\|\cdot\right\|_{\theta},\chi\right)$. Indeed, if $\chi$ is unbounded, then, for all $n>0$ there exists $\ell>0$ and $x,y\in X$ with $d_{1}\left(x,y\right)\leq\theta^{n}$, $\mathbbm{1}_{\{\chi>\ell\}}\left(x\right)=1$, and $\mathbbm{1}_{\{\chi>\ell\}}\left(y\right)=0$. Hence, $\left\|\mathbbm{1}_{\{\chi>\ell\}}\right\|_{\theta}\geq\theta^{-n}$. For this reason it turns out that the quasi-Hölder continuous functions witnessing only one pole are more appropriate observables to allow (7) to hold.

1.2. Main theorem

Before stating our main theorem we first define the notion of regular and slow variation. A function $L$ is called slowly varying if for every $c>0$ we have $L(cx)\sim L(x)$. Here, $u(x)\sim w(x)$ means that $u$ is asymptotic to $w$ at infinity, i.e. $\lim_{x\to\infty}u(x)/w(x)=1$. A function $g$ is called regularly varying with index $\alpha$ if it can be written as $g(x)=L(x)\cdot x^{\alpha}$ with $L$ being slowly varying. For $L$ being slowly varying we denote by $L^{\#}$ a de Bruijn conjugate of $L$, i.e. a slowly varying function satisfying

[TABLE]

For more details see [BGT87, Section 1.5.7 and Appendix 5]. Furthermore, we set

[TABLE]

Theorem 1.3**.**

Let $\left(X,\mathcal{B},\mu,\chi\right)$ fulfill Property $\mathfrak{F}$ and let additionally $\chi$ 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]

See [KS18, Remark 1.8 and Remark 1.9] for remarks on this theorem in the general setting.

Remark 1.4*.*

Note that in [KS18] more strong laws under trimming are provided for tuples $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfilling Property $\mathfrak{D}$ (see Definition 2.3). Our approach is to prove that $\left(X,\mathcal{B},\mu,\chi\right)$ fulfilling Property $\mathfrak{F}$ implies the existence of $\epsilon_{0}\in\left(0,1\right)$ such that $\left(X,\mathcal{B},\sigma,\mu,\mathsf{H},\left\|\cdot\right\|_{\epsilon_{0}},\chi\right)$ fulfills Property $\mathfrak{D}$, see Lemma 3.13. Hence, all the statements given in [KS18] also hold for the tuple $\left(X,\mathcal{B},\sigma,\mu,\mathcal{F},\left\|\cdot\right\|_{\epsilon_{0}},\chi\right)$. For brevity we will not restate these results.

1.3. Trimming statements for Markov systems

Theorem 1.5**.**

Let $X$ be a one-sided subshift of finite type and $\mathbb{A}\coloneqq\left\{\mathfrak{a}_{1},\ldots,\mathfrak{a}_{k}\right\}$, $k\geq 2$, its alphabet. Let $\mu$ be a stationary Markov measure compatible with an irreducible and aperiodic matrix $A\in\left\{0,1\right\}^{k\times k}$ (see Definition 3.1 and Definition 3.3) such that $q\coloneqq\mu\left(x_{n}=\mathfrak{a}_{1}\lvert x_{n-1}=\mathfrak{a}_{1}\right)>0$ and $R\coloneqq\mu\left(x_{1}=\mathfrak{a}_{1}\right)/q$. For $\eta>1/q$ let $\chi:X\to\mathbb{R}_{\geq 0}$ be given by

[TABLE]

Further, let $\left(b_{n}\right)_{n\in\mathbb{N}}$ be a sequence of natural numbers tending to infinity with $b_{n}=o\left(n\right)$ and assume there exists $\psi\in\Psi$ fulfilling

[TABLE]

Then there exists a sequence of constants $(d_{n})$ such that

[TABLE]

Remark 1.6*.*

In certain cases $d_{n}$ can be explicitly given: Assume that $\left(b_{n}\right)$ can be written as $b_{n}=R/(1-q)\cdot q^{k_{n}}\cdot n+w_{n}$ with $w_{n}\geq\left(q^{k_{n}}\cdot n\right)^{1/2+\epsilon}\cdot\log\psi\left(\left\lfloor\log n\right\rfloor\right)^{1/2-\epsilon}$ and $w_{n}=o\left(q^{k_{n}}\right)$ for $\left(k_{n}\right)$ being a sequence of natural numbers, then

[TABLE]

Note that there is an analogy to formula (4) with $\alpha\coloneqq-\log q/\log\eta$ and $L=1$. However, it is not possible to apply the same method here since $\chi$ does not have a regularly varying tail distribution.

Remark 1.7*.*

We will state a theorem with a St. Petersburg type distribution functions also in the more general setting introduced in [KS18], see Theorem 2.4, i.e. in this setting the theorem can also be applied to piecewise expanding interval maps (and possibly other systems).

1.4. Structure of the paper

In Section 2 we state and prove additional limit theorems under the more general setting of Property $\mathfrak{D}$. This is a general property for dynamical systems given in [KS18]. We specify this property in Section 2.1 and state an intermediately trimmed strong law for St. Petersburg type distribution functions in Section 2.2. In Section 2.3 we give a general approach for proving an intermediately trimmed strong law. This is a refinement of the approach given in [KS18, Section 2.1] putting extra emphasize on the appearance of ties. One main ingredient of this proof is a truncated limit theorem given in Section 2.4. Finally, we give the proof of the general theorem in Section 2.5 which will be the basis to prove Theorem 1.5.

Section 3 is devoted to prove certain properties of the space of quasi-Hölder continuous functions for subshifts of finite type. After giving some necessary definitions in Section 3.1 we prove the main property of this space, the spectral gap property, in Section 3.2. In Section 3.3 we prove Theorem 1.3 and Theorem 1.5. The main step is to show that if $(X,\mathcal{B},\mu,\chi)$ fulfills Property $\mathfrak{F}$, then there exists $\epsilon_{0}\in(0,1)$ such that $\left(X,\mathcal{B},\mu,\sigma,\mathsf{H},\left\|\cdot\right\|_{\epsilon_{0}},\chi\right)$ fulfills Property $\mathfrak{D}$. This enables us to use the machinery established in [KS18] and to prove Theorem 1.3. Finally, we take a closer look at the example of Gibbs-Markov measures given in Section 1.3 and show that for this setting Property $\mathfrak{F}$ holds which proves Theorem 1.5.

2. Further limit results in the general setting

2.1. General setting

In the following we denote the spectral radius of an operator $U$ by $\rho\left(U\right)$ and give first the definition of a spectral gap.

Definition 2.1** (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. $PN=NP=0$.

Next we state the two main properties from [KS18]. Under this setting we will prove in the next section an intermediately trimmed strong law for St. Petersburg type distribution functions.

Definition 2.2** (Property $\mathfrak{C}$, [KS18, Definition 1.1]).**

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 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 $\left\|\cdot\right\|$.

The above mentioned property is a widely used setting for dynamical systems.

Definition 2.3** (Property $\mathfrak{D}$, [KS18, Definition 1.2]).**

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 $\mathfrak{C}$.

• •

$\chi\in\mathcal{F}^{+}\coloneqq\{f\in\mathcal{F}\colon f\geq 0\}$.

• •

For $\prescript{\ell}{}{\mathrlap{\smash{\chi}}\phantom{\chi}}\coloneqq\chi\cdot\mathbbm{1}_{\left\{\chi\leq\ell\right\}}$ there exists $K_{1}>0$ such that for all $\ell>0$,

[TABLE]

• •

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

[TABLE]

2.2. Trimming results for St. Petersburg type distribution functions

In this section we will prove theorems under the more general setting that the tuple $\left(\Omega,\mathcal{B},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfills Property $\mathfrak{D}$. From Lemma 3.13 we will see that this can immediately be applied to the setting of subshifts of finite type.

Theorem 2.4**.**

Let $\left(\Omega,\mathcal{B},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$ and assume that there exists $K_{3}>0$ such that for all $\ell>0$

[TABLE]

Additionally let $q\in(0,1)$, $R\in\left(0,1/q\right)$ and $\eta>1/q$ such that for all $k\in\mathbb{N}$ we have that $\mu\left(\chi=\eta^{k}\right)=R\cdot q^{k}$.

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]

If additionally to (9) there exists $\epsilon>0$ such that $\left(b_{n}\right)$ can be written as $b_{n}=R/(1-q)\cdot q^{k_{n}}\cdot n+w_{n}$, where $w_{n}\geq\left(q^{k_{n}}\cdot n\right)^{1/2+\epsilon}\cdot\log\psi\left(\left\lfloor\log n\right\rfloor\right)^{1/2-\epsilon}$ and $w_{n}=o\left(q^{k_{n}}\right)$ and $\left(k_{n}\right)$ is a sequence of natural numbers, then $d_{n}$ can be explicitly given by

[TABLE]

2.3. General approach to the proof of Theorem 2.4

The proof of Theorem 2.4 is similar to the general structure given in [KS18, Lemma 2.3]. However, due to the nature of having ties some additional attention is needed.

The first property considers the sum of truncated random variables. Namely, for $\chi:\Omega\to\mathbb{R}_{\geq 0}$ and for a real valued sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$ we let

[TABLE]

denote the corresponding truncated sum process.

Next, we will give some properties under which an intermediately trimmed strong law can be established.

Definition 2.5** ([KS18, 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 2.6** ([KS18, 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]

A similar property will be given as follows:

Definition 2.7**.**

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

[TABLE]

The following lemma is an extension of [KS18, Lemma 2.3] giving an extra consideration to the appearance of ties.

Lemma 2.8**.**

Let $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$. For the system $\left(\Omega,\mathcal{A},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ let further $(f_{n})$ fulfill Property $\bm{A}$, let $((f_{n}),(\gamma_{n}))$ fulfill Property $\bm{B}$, and let $((f_{n}),(\gamma_{n}^{\prime}))$ fulfill Property $\bm{B^{\prime}}$. For fixed $w>0$ let

[TABLE]

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

[TABLE]

and

[TABLE]

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

[TABLE]

Proof.

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

[TABLE]

Furthermore, Property $\bm{B}$ in conjunction with Property $\bm{B^{\prime}}$ implies that a.s. eventually we have for all

[TABLE]

that $\chi\circ T^{\pi(k)}\geq f_{n}$. This implies that a.s.

[TABLE]

By the restriction of $r_{n}$ in (11) we have

[TABLE]

This implies that we have a.s.

[TABLE]

On the other hand, since ${{}^{f_{n}}}\chi\leq f_{n}$ it follows by Property $\bm{B}$ that a.s.

[TABLE]

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

[TABLE]

Using (12) and (13) together with the range of $r_{n}$ given in (11) yields

[TABLE]

Combining this with Property $\bm{A}$ and (16) gives the statement of the lemma. ∎

In the next section we give a statement under which conditions Property $\bm{A}$ holds.

2.4. Truncated random variables for St. Petersburg type distribution functions

We will first state a strong limit law for the truncated sum $\mathsf{T}_{n}^{f_{n}}\chi$.

Theorem 2.9**.**

Let $\left(\Omega,\mathcal{B},T,\mu,\mathcal{F},\left\|\cdot\right\|,\chi\right)$ fulfill Property $\mathfrak{D}$. For given $\eta>1$ assume $\mu\left(\chi=\eta^{k}\right)=R\cdot q^{k}$ with $q\in\left(1/\eta,1\right)$ and $R<1/q$, for all $k\in\mathbb{N}$. Let $\left(f_{n}\right)_{n\in\mathbb{N}}$ be a positive valued sequence with $F\left(f_{n}\right)>0$, for all $n\in\mathbb{N}$, and $\lim_{n\to\infty}f_{n}=\infty$. If there exists $\psi\in\Psi$ such that

[TABLE]

then

[TABLE]

Remark 2.10*.*

A similar setting was also studied in [GK11] and [Nak15]. Particularly, [Nak15, Ex. 1.1] considers the same distribution function for the i.i.d. setting with $q$ restricted to $1/2$. A combination of Theorem 1.2 and Corollary 1.1 of [Nak15] gives the limit result as in (18) but imposes a stronger condition on $(f_{n})$ than (17).

Next we give some technical results which will help us to prove Theorem 2.9 and Theorem 2.4.

Lemma 2.11**.**

Assume that a sequence $(f_{n})$ can be written as $f_{n}=\eta^{k_{n}}$ with $(k_{n})$ a sequence of natural numbers. Then

[TABLE]

If additionally $f_{n}$ tends to infinity, then

[TABLE]

Proof.

As $f_{n}=\eta^{\log f_{n}/\log\eta}$ (19) immediately follows. This also gives

[TABLE]

i.e. (20) follows. Finally,

[TABLE]

giving (21). ∎

In order to prove Theorem 2.9 we will make use of the following Lemma.

Lemma 2.12** ([KS18, 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 $\epsilon\in\left(0,E\right)$, $r>0$ with $F(r)>0$, and $n\in\mathbb{N}_{>N^{\prime}}$

[TABLE]

(The lemma is stated slightly different in [KS18] but the statement in the current version becomes obvious from the proof of the lemma in [KS18].)

Proof of Theorem 2.9.

The proof is very similar to [KS18, Proof of Theorem 2.5], but as $F$ is not regularly varying the methods can not immediately be transferred.

We define the sequences $\left(g_{n}\right)_{n\in\mathbb{N}}$ and $\left(\overline{g}_{n}\right)_{n\in\mathbb{N}}$ with $g_{n}\coloneqq\max\left\{f_{n},n^{1/2}\right\}$ and $\overline{g}_{n}\coloneqq\min\left\{f_{n},n^{1/2}\right\}$ and prove separately that

[TABLE]

and

[TABLE]

hold.

To prove (22) we set $I_{j}\coloneqq\left[2^{j},2^{j+1}-1\right]$ for $j\in\mathbb{N}$ and obtain for every $m\in I_{j}$ and $r\in\mathbb{R}_{\geq 0}$ that

[TABLE]

Next we define the sequences $\left(s_{j}\right)_{j\in\mathbb{N}}$ and $\left(t_{j}\right)_{j\in\mathbb{N}}$ as

[TABLE]

These numbers are chosen such that $\left[\eta^{s_{j}},\eta^{t_{j}}\right]\supset\left[\min_{n\in I_{j}}g_{n},\max_{n\in I_{j}}g_{n}\right]$. Furthermore, we know that $f_{n}\in[\eta^{k},\eta^{k+1})$ implies $\mathsf{T}_{n}^{f_{n}}\chi=\mathsf{T}_{n}^{\eta^{k}}\chi$. Hence, (24) implies

[TABLE]

for all $j\in\mathbb{N}$. In order to estimate the sets on the right hand side of (25) we can apply Lemma 2.12 assuming that $\epsilon/2<E$ to the sum $\mathsf{T}_{n}^{\eta^{k}}\chi$ and obtain for $j$ sufficiently large

[TABLE]

Furthermore, (21) implies

[TABLE]

Here, we write $a_{\ell}\asymp b_{\ell}$ if there exists a constant $c>0$ such that $c^{-1}a_{\ell}\leq b_{\ell}\leq ca_{\ell}$ for all $\ell\in\mathbb{N}$ and in (27) $\asymp$ holds both with respect to $k$ and $j$. Hence, we obtain by (26) that there exists $W>0$ such that

[TABLE]

for all $k\in\mathbb{N}$ and $j$ sufficiently large. This implies

[TABLE]

To continue we state the following technical lemma which is [KS17, Lemma 5].

Lemma 2.13**.**

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

[TABLE]

Noticing that

[TABLE]

and using condition (17) together with Lemma 2.13 implies that there exists $\psi\in\Psi$ such that

[TABLE]

Inserting this into the calculation in (28) yields

[TABLE]

for $j$ sufficiently large. Using (25), summing over the above quantity with respect to $j$, using the fact that $\psi\in\Psi$, and applying the Borel-Cantelli lemma yields (22).

On the other hand using Lemma 2.12 again yields

[TABLE]

for $\epsilon\in\left(0,E\right)$ and $n$ sufficiently large. Summing over the above quantity and using the Borel-Cantelli lemma yields that for all $\epsilon>0$ (23) holds giving the statement of the theorem. ∎

2.5. Proof of Theorem 2.4

We start with two lemmas regarding Properties $\bm{B}$ and $\bm{B^{\prime}}$. Before stating them we define

[TABLE]

for $k\in\mathbb{R}_{\geq 1}$, $n\in\mathbb{N}_{\geq 3}$, $0<\epsilon<1/4$, and $\psi\in\Psi$.

Lemma 2.14** ([KS18, Lemma 2.8]).**

Let $\left(u_{n}\right)_{n\in\mathbb{N}}$ be a positive valued sequence and define $p_{n}\coloneqq\mu\left(\chi>u_{n}\right)$. Then there exist constants $N,V>0$ such that for all $0<\epsilon<1/4$, $\psi\in\Psi$, $n\in\mathbb{N}_{>N}$, and $\left(u_{n}\right)$ positive valued we have

[TABLE]

Lemma 2.15**.**

Let $\left(v_{n}\right)_{n\in\mathbb{N}}$ be a positive valued sequence and define $p_{n}^{\prime}\coloneqq\mu\left(\chi=v_{n}\right)$. Then there exist constants $N^{\prime},V^{\prime}>0$ such that for all $0<\epsilon<1/4$, $\psi\in\Psi$, $n\in\mathbb{N}_{>N^{\prime}}$, and $\left(v_{n}\right)$ positive valued we have

[TABLE]

As the proof of Lemma 2.15 is mainly the same as the proof of Lemma 2.14 given in [KS18], we will not repeat it here. The only difference is that we use (8) instead of (7).

With those two properties at hand we are able to prove Theorem 2.4.

Proof of Theorem 2.4.

In the first part of the proof we will show (3) using Lemma 2.8. We define

[TABLE]

with $c$ as in (29) and $V$ given in Lemma 2.14. With this choice of $(f_{n})$ we have that

[TABLE]

We first want to show the following: Let $(b_{n})$ be as in (9) and set

[TABLE]

for all $n\in\mathbb{N}$. Then there exists $w\in\mathbb{R}_{>0}$ such that for all $n\in\mathbb{N}$ the sequence $(b_{n})$ can be written as

[TABLE]

with

[TABLE]

This can be seen as follows: As a first boundary case set $b_{n}^{\prime}\coloneqq\left\lceil n\cdot\mu(\chi>f_{n})+\gamma_{n}\right\rceil$, then

[TABLE]

As the second boundary case set

[TABLE]

Assume that $\left(b_{n}\right)=\left(b_{n}^{\prime\prime}\right)$ fulfills (9) for some $\psi\in\Psi$. If we consider $c_{\epsilon,\psi}\left(k,n\right)$ in (29) for the same $\psi$, then $c\left(b_{n}^{\prime\prime},n\right)=o\left(b_{n}^{\prime\prime}\right)$. Since $n\cdot\mu\left(\chi=f_{n}\right)\leq b_{n}^{\prime\prime}$ and $c$ is monotonically increasing in its first argument, we also have that $w\cdot\left(\gamma_{n}+\gamma_{n}^{\prime}\right)=o\left(b_{n}^{\prime\prime}\right)$. Applying (19) and (20) gives

[TABLE]

which implies that there exists $w>0$ such that $V\cdot c\left(b_{n}^{\prime\prime},n\right)<w\cdot\left(\gamma_{n}+\gamma_{n}^{\prime}\right)$, for all $n\in\mathbb{N}$. Hence,

[TABLE]

Finally, (33) and (35) imply $b_{n}\in\left[b_{n}^{\prime},b_{n}^{\prime\prime}\right]$ and (32) follows.

Furthermore, by Lemma 2.14 and Lemma 2.15 we have that the pair $((f_{n}),(\gamma_{n}))$ fulfills Property $\bm{B}$ and $((f_{n}),(\gamma_{n}^{\prime}))$ fulfills Property $\bm{B^{\prime}}$ for $(\gamma_{n})$ and $(\gamma_{n}^{\prime})$ being defined as in (31).

As in (34) we can show that $b_{n}\asymp n\cdot\mu\left(\chi=f_{n}\right)$. This observation combined with (9) and (19) implies

[TABLE]

which is equivalent to (17) and Theorem 2.9 states that under this condition Property $\bm{A}$ holds.

From (19) and (21) it immediately follows that (12) holds.

Finally, we will prove (13). Since $\mu(\chi=f_{n})\asymp\mu(\chi>f_{n})$ by (19) and (20), we have by the definition of $c$ that also $\gamma_{n}\asymp\gamma_{n}^{\prime}$ holds. Using first (21) and then (40) gives

[TABLE]

Since $b_{n}\asymp n\cdot\mu(\chi=f_{n})$ and $b_{n}$ fulfills (9), we have that $c(n\cdot\mu(\chi=f_{n}),n)=o(n\cdot\mu(\chi=f_{n}))$. Thus, $\gamma_{n}=o\left(n\cdot\mu\left(\chi=f_{n}\right)\right)$ implying (13).

Hence, we can apply Lemma 2.8 and obtain the first part of the theorem.

Next we show the asymptotic given in (10). The first step is to prove that $f_{n}=\eta^{k_{n}}$ for the definition of $f_{n}$ in (30). First note that $w_{n}\geq\left(q^{k_{n}}\cdot n\right)^{1/2+\epsilon}\cdot\log\psi\left(\left\lfloor\log n\right\rfloor\right)^{1/2-\epsilon}$ together with (9) implies $w_{n}\geq c_{\epsilon/2,\psi}\left(R/\left(1-q\right)\cdot q^{k_{n}}\cdot n,n\right)$, for $n$ sufficiently large. As $w_{n}=o\left(q^{k_{n}}\cdot n\right)$ this also implies $w_{n}\geq V\cdot c_{\epsilon/2,\psi}\left(b_{n},n\right)$, for $n$ sufficiently large. As we can choose $\epsilon$ arbitrarily for $c_{\epsilon,\psi}$ in (30), $f_{n}$ can be set such that $n\cdot R/\left(1-q\right)\cdot q^{k_{n}}\geq n\cdot\mu\left(\chi>f_{n}\right)$. This together with (20) implies $f_{n}\leq\eta^{k_{n}}$.

On the other hand, as $w_{n}=o\left(n\cdot q^{k_{n}}\right)$ we also have that

[TABLE]

implying $f_{n}>\eta^{k_{n}-1}$ and thus $f_{n}=\eta^{k_{n}}$. In particular, it follows that $n\cdot\mu(\chi>f_{n})\sim b_{n}$.

Using (20) yields

[TABLE]

Hence, using (21) gives

[TABLE]

∎

3. Properties of the space of quasi-Hölder continuous functions

3.1. Further definitions and remarks

We first give some standard definitions which we have omitted in the introduction.

Definition 3.1**.**

A matrix $A\in\left\{0,1\right\}^{k\times k}$ is called irreducible if for each pair $\left(i,j\right)$ with $1\leq i,j\leq k$, there exists an $n\in\mathbb{N}$ such that $A^{n}\left(i,j\right)>0$. We define the period $d$ of $A$ as

[TABLE]

where $\gcd$ denotes the greatest common divisor. The matrix $A$ is called aperiodic if $d=1$.

Definition 3.2**.**

A probability measure $\mu$ on $X$ is called Gibbs measure if there exist a measurable function $f\colon X\to\mathbb{R}$ and constants $K>0$ and $C\in\mathbb{R}$ such that for all $n\in\mathbb{N}$, all $\mathsf{A}\in\mathbb{A}_{n}$, and all $x\in[\mathsf{A}]$ we have that

[TABLE]

Definition 3.3**.**

Let $\mathbb{A}=\left\{\mathfrak{a}_{1},\ldots,\mathfrak{a}_{k}\right\}$, $k\geq 2$ be a finite alphabet. Further, let $A\in\left\{0,1\right\}^{k\times k}$ and $P\in\left[0,1\right]^{k\times k}$ with $p_{i,j}=0$ if and only if $a_{i,j}=0$ and $\sum_{j=1}^{k}p_{i,j}=1$, for all $1\leq i\leq n$. Further, let $\pi$ be an $n$-vector such that $\pi\cdot P=\pi$. If a measure $\mu$ is defined on cylinder sets as

[TABLE]

then it is called Markov measure corresponding to $A$.

Next we recall the definition of $g$-functions and $g$-measures.

Definition 3.4**.**

A measurable function $g\colon X\to\mathbb{R}_{\geq 0}$ is called $g$-function if

[TABLE]

for all $y\in X$. If $g=\mathrm{d}\mu/\mathrm{d}\mu\circ\sigma$, then $\mu$ is called $g$-measure for a $g$-function $g$.

For the following let $L_{f}\colon\mathcal{L}^{1}\to\mathcal{L}^{1}$ denote the Perron-Frobenius operator given by

[TABLE]

If $\exp(f)$ is a $g$-function, then $L_{f}$ is normalized, i.e. $L_{f}1=1$.

Proposition 3.5**.**

Let $f\colon X\to X$ be Lipschitz continuous with respect to the metric $d_{1}$ such that $\exp(f)$ is a $g$-function. Then there exists a corresponding $g$-measure $\mu$. This measure is a $\sigma$-invariant Gibbs measure with constant $C=0$ and as such mixing. Moreover, the Perron-Frobenius operator $L_{f}$ coincides with the definition of the transfer operator given in (5).

This proposition follows from a direct application of [Kea72, Theorem on p. 134], [PP90, Corollary 3.2.1] and [Wal75, Corollary 3.3]. The last statement follows by direct calculation.

3.2. The spectral gap property

To show that the transfer operator $\widehat{\sigma}$ has a spectral gap on $\mathsf{H}$ we first have to ensure that $\mathsf{H}$ is a Banach space.

Lemma 3.6**.**

$\mathsf{H}$* is a Banach algebra containing the constant functions and fulfills $\left|\cdot\right|_{\infty}\leq\left\|\cdot\right\|_{\epsilon_{0}}$, for each $\epsilon_{0}\leq 1$.*

Proof.

Let $f,g\in\mathsf{H}$. Since

[TABLE]

and all other properties of a norm follow immediately, we find that $\left\|\cdot\right\|_{\epsilon_{0}}$ is a norm.

In the next steps we will show completeness following the proof in [Bla97, Lemma 2.3.17]. Let $\left(f_{n}\right)$ be a Cauchy sequence with respect to $\left\|\cdot\right\|_{\epsilon_{0}}$. In particular, $\left(f_{n}\right)$ is also a Cauchy sequence with respect to $\left|\cdot\right|_{\infty}$, we set $f$ as its limit. So our next step is to prove that $f\in\mathsf{H}$. Since $\left(f_{n}\right)$ is a Cauchy sequence with respect to $\left\|\cdot\right\|_{\epsilon_{0}}$, for each $\delta>0$ we can choose $L>0$ such that $\left\|f_{k}-f_{\ell}\right\|_{\epsilon_{0}}<\delta$ for all $k,\ell>L$. Then we have that

[TABLE]

By Fatou’s lemma we have that the limit $\ell\to\infty$ on the right hand side exists and thus,

[TABLE]

Thus, $f\in\mathsf{H}$ and $\left(f_{n}\right)$ converges to $f$ with respect to $\left\|\cdot\right\|_{\epsilon_{0}}$.

We further note that for all $f,g\in\mathsf{H}$

[TABLE]

and thus,

[TABLE]

Hence, $\left(\mathsf{H},\left\|\cdot\right\|_{\epsilon_{0}}\right)$ is a Banach algebra, the constant functions are obviously contained, and $\left|\cdot\right|_{\infty}\leq\left\|\cdot\right\|_{\epsilon_{0}}$ is clear from the definition of $\left\|\cdot\right\|_{\epsilon_{0}}$. ∎

To prove that $\widehat{\sigma}$ has a spectral gap, we use the following theorem by Hennion and Hervé which is a generalization of a theorem by Doeblin and Fortet, [DF37], and Ionescu–Tulcea and Marinescu, [ITM50].

Lemma 3.7** ([HH01, Theorem II.5]).**

Suppose $\left(\mathcal{L},\left\|\cdot\right\|\right)$ is a Banach space and $U:\mathcal{L}\to\mathcal{L}$ is a bounded linear operator with spectral radius $1$. Assume that there exists a semi-norm $\left|\cdot\right|^{\prime}$ with the following properties:

1. (a)

$\left|\cdot\right|^{\prime}$* is continuous on $\mathcal{L}$.* 2. (b)

$U$* is bounded on $\mathcal{L}$ with respect to $\left|\cdot\right|^{\prime}$, i.e. there exists $M>0$ such that $\left|Uf\right|^{\prime}\leq M\left|f\right|^{\prime}$, for all $f\in\mathcal{L}$.* 3. (c)

There exist constants $0<r<1$, $R>0$, and $n_{0}\in\mathbb{N}$ such that

[TABLE]

for all $f\in\mathcal{L}$. 4. (d)

$U\left\{f\in\mathcal{L}\colon\left|f\right|^{\prime}<1\right\}$* is precompact on $\left(\mathcal{L},\left\|\cdot\right\|\right)$, i.e. for each sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$ with values in $\mathcal{L}$ fulfilling $\sup_{n\in\mathbb{N}}\left|f_{n}\right|^{\prime}<1$ there exists a subsequence $\left(n_{k}\right)$ and $g\in\mathcal{L}$ such that*

[TABLE]

Then $U$ is quasi-compact, i.e. there is a direct sum decomposition $\mathcal{L}=F\oplus H$ and $0<\tau<\rho\left(U\right)$ where

• •

$F$, $H$ are closed and $U$-invariant, i.e. $U\left(F\right)\subset F$, $U\left(H\right)\subset H$,

• •

$\dim\left(F\right)<\infty$* and all eigenvalues of $U\lvert_{F}:\mathcal{F}\to\mathcal{F}$ have modulus larger than $\tau$, and*

• •

$\rho\left(U\lvert_{H}\right)<\tau$.

With the following lemma we will show that $\widehat{\sigma}$ has a spectral gap. It is a standard approach, but for completeness we will also give a proof of this lemma.

Lemma 3.8**.**

If $U$ is quasi-compact, $U$ has a unique eigenvalue on $\left\{z\colon\left|z\right|=\rho\left(U\right)\right\}$, and this eigenvalue is simple, then $U$ has a spectral gap.

Proof.

We use the decomposition of $\mathcal{L}$ from Lemma 3.7. Since $F$ is finite dimensional, we can calculate the Jordan form of $L\lvert_{F}$. Since all eigenvalues of $L\lvert_{F}$ have modulus larger than $\tau$, the unique and simple eigenvalue on $\left\{z\colon\left|z\right|=\rho\left(U\right)\right\}$ is also unique and simple in $L\lvert_{F}$. Hence, the Jordan form consists of a $1\times 1$- block with eigenvalue $\lambda$ such that $\left|\lambda\right|=\rho\left(U\right)$, and possibly other Jordan blocks with eigenvalues $\lambda_{i}$ such that $\left|\lambda_{i}\right|<\left|\lambda\right|$ for each $i$. Hence, $F$ can be decomposed into $F=\operatorname{span}\left\{\nu\right\}\oplus F^{\prime}$, where $U\nu=\lambda\nu$ and $\rho\left(U\lvert_{F^{\prime}}\right)<\left|\lambda\right|$. Thus,

[TABLE]

where $U\left(H^{\prime}\right)\subset H^{\prime}$ and $\rho\left(U\lvert_{H^{\prime}}\right)<\left|\lambda\right|$.

For the following we define $\Pi_{1}$ and $\Pi_{2}$ as the unique projections to the spaces $\operatorname{span}\left\{\nu\right\}$ and $H^{\prime}$, i.e. for every $f\in\mathcal{L}$ it holds $f=\Pi_{1}f+\Pi_{2}f$. These projections are idempotent, i.e. $\Pi_{1}^{2}=\Pi_{1}$ and $\Pi_{2}^{2}=\Pi_{2}$. Since $U\left(\operatorname{span}\left\{\nu\right\}\right)\subset\operatorname{span}\left\{\nu\right\}$ and $U\left(H^{\prime}\right)\subset H^{\prime}$, we have that

[TABLE]

and

[TABLE]

Since $\dim\left(\operatorname{span}\left\{\nu\right\}\right)=1$, we have that $\dim\left(\operatorname{Im}\left(P\right)\right)=1$ and $P$ is obviously a projection. Furthermore, we have

[TABLE]

and analogously $NP=0$, which finishes the proof the lemma. ∎

Lemma 3.9**.**

Let $\widehat{\sigma}$ be defined as in (5), then $\widehat{\sigma}$ has a simple eigenvalue $\lambda=\rho\left(\widehat{\sigma}\right)=1$. This eigenvalue is unique on the unit circle and has maximal modulus.

The proof is standard, but see also [KS18, Lemma 3.2]. For the next lemma let $L_{f}$ denote the Perron-Frobenius operator given in (36).

Proposition 3.10**.**

For $f\in F_{\theta}$, there exists $\epsilon_{0}\in\left(0,1\right)$ such that $L_{f}$ is a bounded linear operator on $\mathsf{H}$ with respect to $\left\|\cdot\right\|_{\epsilon_{0}}$ and has a spectral gap.

Proof.

We aim to apply Lemma 3.7 in combination with Lemma 3.8 and Lemma 3.9 to the space $\left(\mathsf{H},\left\|\cdot\right\|_{\epsilon_{0}}\right)$ with $\left|\cdot\right|_{1}$ as the semi-norm $\left|\cdot\right|^{\prime}$. So we only have to show that $\left|\cdot\right|_{1}$ fulfills (a) to (d).

ad (a): Obviously, $\left|\cdot\right|_{1}$ is continuous.

ad (b): Since $\mathrm{e}^{f}$ is positive and $L_{f}$ is normalized and the integral is $L_{f}$-invariant, it follows that

[TABLE]

i.e.  $L_{f}$ is bounded on $\mathsf{H}$ with respect to $\left|\cdot\right|_{1}$.

ad (c): Before we can start with the proof of (c) we need the following two lemmas:

Lemma 3.11**.**

There exists $\gamma\in\left(0,1\right)$, $K>0$ and $u\in\mathbb{N}$ such that for all $n\in\mathbb{N}$ and all $\mathsf{A}\in\mathbb{A}_{n}$

[TABLE]

Proof.

By the Gibbs property, see Proposition 3.5, there exists $K>0$ such that for all $n\in\mathbb{N}$, $\mathsf{A}\in\mathbb{A}_{n}$, and $x\in\left[\mathsf{A}\right]$ we have that

[TABLE]

Hence,

[TABLE]

To prove (38) we use that $A$ is irreducible and aperiodic which implies that there is a maximal number $u$ such that all $x,\sigma(x),\ldots,\sigma^{u-2}(x)$ have only one preimage, i.e. in this case $\sigma^{-1}(\sigma^{u-1}(x))$ consists of at least two points.

On the other hand, if $x$ has more than one preimage, then $\exp(f(y))<1$ for all $y$ fulfilling $\sigma(y)=x$. This implies

[TABLE]

As $\exp(f)$ is bounded from below and $\sum_{\sigma(y)=x}\exp(f(y))=1$ the above term has to be less than one. Applying (39) and setting $\gamma=\sup_{\{z\in X\colon\exp(f(z))<1\}}\exp(f(z))$ gives the statement of (38). ∎

Lemma 3.12**.**

There exist $\epsilon_{0},s\in(0,1)$ and $L\in\mathbb{N}$ such that for all $\ell\geq L$, $\mathsf{A}\in\mathbb{A}_{\ell}$, $x\in X$ with $\mathsf{A}x$ admissible, and $\epsilon\in(0,\epsilon_{0}]$ we have that $[\mathsf{A}]\cap\sigma^{-\ell}(B(\epsilon,x))\subset B(s\epsilon,\mathsf{A}x)$.

Proof.

For all $\epsilon>0$ and $x\in X$ there exists $n\in\mathbb{N}$ such that $B(\epsilon,x)=[x_{1}\ldots x_{n}]$. Hence, if we write $[\mathsf{A}]=\mathsf{a}_{1}\ldots\mathsf{a}_{\ell}$, then $[\mathsf{A}]\cap\sigma^{-\ell}(B(\epsilon,x))=[\mathsf{a}_{1}\ldots\mathsf{a}_{\ell}x_{1}\ldots x_{n}]$. By the Gibbs property from Proposition 3.5 we have

[TABLE]

Furthermore, Lemma 3.11 implies the existence of $L\in\mathbb{N}$ and $s\in(0,1)$ such that we have for all $\ell\geq L$ and all $\mathsf{A}\in\mathbb{A}_{\ell}$ that $K^{3}\cdot\mu\left(\left[\mathsf{a}_{1}\ldots\mathsf{a}_{\ell}\right]\right)\leq s<1$. Combining this consideration with (40) yields the statement of the lemma. ∎

Now we are in the position to begin with the proof of (c). We have that

[TABLE]

Since $\epsilon_{0}<1$ we have that $B\left(\epsilon,x\right)$ is a cylinder set of length at least one and we can conclude that

[TABLE]

For the following calculations we assume that for given $\mathsf{A}\in\mathbb{A}_{\ell}$ and $x\in X$ we have that $\mathsf{A}x$ is admissible and thus $\left[\mathsf{A}\right]\cap\sigma^{-\ell}\left(B\left(\epsilon,x\right)\right)\neq\varnothing$. Using [Sau00, Proposition 3.2 (iii)] which can also be applied to the situation here yields

[TABLE]

In order to estimate the first summand we notice that by Lemma 3.12 there exists $s<1$ such that for all $\mathsf{A}\in\mathbb{A}_{\ell}$ we have that

[TABLE]

Furthermore, (38) implies that for all $\epsilon\in\left(0,1\right)$ there exists $n_{\epsilon}\in\mathbb{N}$ such that for all $x\in X$ the set $B\left(\epsilon,x\right)$ is a cylinder of length at least $n_{\epsilon}$. Then $\mathsf{A}\in\mathbb{A}_{\ell}$ implies that $\left[\mathsf{A}\right]\cap\sigma^{-\ell}\left(B\left(\epsilon,x\right)\right)$ is a cylinder set of length at least $n+\ell$. Let $R$ be the Lipschitz constant of $f$. Then we have that

[TABLE]

Noting that $\mathrm{e}^{\mathsf{S}_{\ell}f(x)}>0$ for all $x\in X$ yields for the second summand of (42) that

[TABLE]

where the last line follows from (44).

Combining (41), (42), (43), (44), and (45) yields

[TABLE]

Integrating with respect to $\mu$ and noting that the integral is $L_{f}$-invariant yields

[TABLE]

Using the definition of $\left|w\right|_{\epsilon_{0}}$ and assuming that $\epsilon<\epsilon_{0}$ yields

[TABLE]

If $n_{\epsilon}$ is large enough, we have that

[TABLE]

We choose $\epsilon_{0}$ sufficiently small such that (46) is fulfilled for $\epsilon\coloneqq\epsilon_{0}$ and set $n_{0}\coloneqq n_{\epsilon_{0}}$ and $\xi\coloneqq s\cdot\exp\left(R\cdot\theta^{n_{0}}/\left(1-\theta\right)\right)$. Using again the definition of $\left|w\right|_{\epsilon_{0}}$ gives

[TABLE]

and thus the Hennion-Hervé inequality (37) follows.

ad (d): We first prove that $\mathsf{K}\coloneqq\{h:\left\|h\right\|_{\epsilon_{0}}<1\}$ is compact using the approach of [Bla97, Lemma 2.3.18]. From (38) we can conclude that for all $\epsilon>0$ there exists $M\in\mathbb{N}$ such that for all $m\geq M$ and $\mathsf{A}\in\mathbb{A}_{m}$ we have $\mu\left(\left[\mathsf{A}\right]\right)\leq\epsilon$.

We define $\mathcal{B}_{m}$ as the sigma algebra generated by the cylinder sets of length $m$. We note that the conditional expectations $\mathbb{E}\left(h|\mathcal{B}_{m}\right)$ are in particular piecewise constant functions. For $\epsilon\leq\epsilon_{0}$ we have

[TABLE]

The last inequality follows from the fact that for each cylinder set $C$ we have that $\operatorname*{esssup}_{x\in C}h(x)\geq\operatorname*{esssup}_{x\in C}\mathbb{E}\left(h|\mathcal{B}_{m}\right)\left(x\right)$ and $\operatorname*{essinf}_{x\in C}h(x)\leq\operatorname*{essinf}_{x\in C}\mathbb{E}\left(h|\mathcal{B}_{m}\right)\left(x\right)$ which yields $\left|\mathbb{E}\left(h|\mathcal{B}_{m}\right)\right|_{\epsilon_{0}}\leq\left|h\right|_{\epsilon_{0}}\leq 1$, since $h\in\mathsf{K}$.

In the following fix an arbitrary sequence $\left(h_{n}\right)\subset\mathsf{K}$ and a new sequence of function $(h_{n}^{\left(m\right)})_{n\in\mathbb{N}}\coloneqq\left(\mathbb{E}\left(h_{n}|\mathcal{B}_{m}\right)\right)_{n\in\mathbb{N}}$. For given $m\in\mathbb{N}$ we know that $(h_{n}^{\left(m\right)})$ is a sequence of bounded functions being piecewise constant on the same finite number of intervals. Hence, there exists a subsequence $n\left(j,m\right)$ such that $(h_{n\left(j,m\right)}^{\left(m\right)})_{j\in\mathbb{N}}$ is a Cauchy sequence in $\mathcal{L}^{1}$ and thus converges. Additionally we might require the function $n:\mathbb{N}^{2}\to\mathbb{N}$ being such that for each $m\in\mathbb{N}$ we have $\left\{n\left(j,m+1\right)\colon j\geq 1\right\}\subset\left\{n\left(j,m\right)\colon j\geq 1\right\}$. If we set $\overline{n}\left(j\right)\coloneqq n\left(j,j\right)$, then for each $m\in\mathbb{N}$ the sequence $(h_{\overline{n}\left(j\right)}^{\left(m\right)})_{j\in\mathbb{N}}$ is a Cauchy sequence in $\mathcal{L}^{1}$. We can conclude that for all $\epsilon>0$ and $m_{0}\in\mathbb{N}$ there exists $J\in\mathbb{N}$ such that for all $m\geq m_{0}$ and all $j,\ell\geq\max\left\{m_{0},J\right\}$ we have that

[TABLE]

are fulfilled at the same time.

In the last steps we will apply (47) and (48) to obtain

[TABLE]

which proves that $\left(h_{\overline{n}\left(j\right)}\right)$ is a Cauchy sequence and thus convergent in $\mathcal{L}^{1}$. Hence, each sequence has a convergent subsequence and $\mathsf{K}$ is compact.

Since $\mathsf{K}$ is compact, we consider in the following an arbitrary sequence $\left(h_{n}\right)$ with $\left\|h_{n}\right\|_{\epsilon_{0}}<1$, for all $n\in\mathbb{N}$. Since $\mathsf{H}$ is complete, see Lemma 3.6, there exists a subsequence $n_{j}$ such that $h_{n_{j}}$ is Cauchy and there also exists $h\in\mathsf{H}$ with $\left\|h\right\|_{\epsilon_{0}}<1$ such that $\lim_{j\to\infty}\left\|h_{n_{j}}-h\right\|_{\epsilon_{0}}=0$. By the proof of (b) we have that $\left|L_{f}h_{n_{j}}-L_{f}h\right|_{1}=\left|L_{f}\left(h_{n_{j}}-h\right)\right|_{1}\leq\left|h_{n_{j}}-h\right|_{1}$ which tends to zero. Setting $g\coloneqq L_{f}h$ yields (d).

Having proved (a) to (d) we can apply Lemma 3.7 and obtain that $L_{f}$ is quasi-compact. Combining this with Lemma 3.9 and Lemma 3.8 yields that $L_{f}$ has a spectral gap. ∎

3.3. Proofs of Theorems 1.3 and 1.5

We will first prove the following key lemma:

Lemma 3.13**.**

Assume that $(X,\mathcal{B},\mu,\chi)$ fulfills Property $\mathfrak{F}$. Then there exists $\epsilon_{0}\in(0,1)$ such that $\left(X,\mathcal{B},\mu,\sigma,\mathsf{H},\left\|\cdot\right\|_{\epsilon_{0}},\chi\right)$ fulfills Property $\mathfrak{D}$.

Proof.

By Proposition 3.5, $\mu$ is mixing and $\sigma$-invariant. By Lemma 3.6 $\mathsf{H}$ is a Banach algebra of functions which contains the constant functions and fulfills $\left\|\cdot\right\|_{\epsilon_{0}}\geq\left|\cdot\right|_{\infty}$. Proposition 3.10 implies that there exists $\epsilon_{0}>0$ such that $\widehat{\sigma}$ is a bounded linear operator with respect to $\left\|\cdot\right\|_{\epsilon_{0}}$ and has spectral gap on $\mathsf{H}$. Hence, $\left(X,\mathcal{B},T,\mu,\mathsf{H},\left\|\cdot\right\|_{\epsilon_{0}}\right)$ fulfills Property $\mathfrak{C}$. Additionally, (1) and (2) imply that (6) and (7) are fulfilled for the same $\epsilon_{0}$ giving Property $\mathfrak{D}$. ∎

Proof of Theorem 1.3.

Theorem 1.3 follows immediately by applying Lemma 3.13 on [KS18, Theorem 1.7]. This theorem gives the same statement as Theorem 1.3 with the condition on $\left(X,\mathcal{B},\mu,\chi\right)$ fulfilling Property $\mathfrak{F}$ being replaced by the condition of $\left(X,\mathcal{B},\mu,\sigma,\mathsf{H},\left\|\cdot\right\|_{\epsilon_{0}},\chi\right)$ fulfilling Property $\mathfrak{D}$. ∎

Proof of Theorem 1.5.

We have to show that $(X,\mathcal{B},\mu,\chi)$ fulfills Property $\mathfrak{F}$. Applying then Lemma 3.13 and Theorem 2.4 immediately implies the statement of Theorem 1.5.

One can easily calculate that $\mu$ is a $g$-measure and $g\colon X\to X$ with $g\left(y\right)=\mu\left(x_{1}=y_{1}|x_{2}=y_{2}\right)$ the corresponding $g$-function and it follows immediately that $\log g$ is Lipschitz continuous.

It remains to show (1) and (2). Let $[\mathfrak{a}_{1}]_{n}$ denote the $n$-cylinder $\left[\mathfrak{a}_{1}\ldots\mathfrak{a}_{1}\right]$. For given $\epsilon\in(0,1)$ there exists $n\in\mathbb{N}_{0}$ such that $\epsilon\in\left[\mu([\mathfrak{a}_{1}]_{n}),\mu([\mathfrak{a}_{1}]_{n-1})\right)$. This implies

[TABLE]

This implies

[TABLE]

Since the choice of $\epsilon_{0}$ was arbitrary, (1) holds. Similarly, we have for the same choice of $\epsilon$

[TABLE]

This implies

[TABLE]

implying (2) since $\epsilon_{0}^{\prime}$ was arbitrary. ∎