Weak invariance principle in Besov spaces for stationary martingale differences

TL;DR

This paper extends the classical Donsker weak invariance principle to Besov spaces, analyzing polygonal line processes from stationary martingale differences and i.i.d. variables, demonstrating the optimality of the results.

Contribution

It introduces a new invariance principle in Besov spaces for stationary martingale differences, expanding the theoretical framework beyond classical settings.

Findings

Extension of Donsker's principle to Besov spaces

Optimality of the derived results

Applicability to stationary martingale differences and i.i.d. variables

Abstract

The classical Donsker weak invariance principle is extended to a Besov spaces framework. Polygonal line processes build from partial sums of stationary martingale differences as well independent and identically distributed random variables are considered. The results obtained are shown to be optimal.

Equations

$$S_{f,0}:=0,S_{f,n}:=\sum _{j=0}^{n-1}f\circ {\mathsf{T}}^j,n⩾1.$$

$${\zeta }_{f,n}(t):=S_{f,\lfloor nt\rfloor }+(nt-\lfloor nt\rfloor )f\circ {\mathsf{T}}^{\lfloor nt\rfloor },$$

$$\Vert x{\Vert }_p={\left({\int }_0^1{\left|x(t)\right|}^p\mathrm{d}t\right)}^{1/p},x\in L_p([0,1]).$$

$${\omega }_p(x,\delta )=\underset{|h|⩽\delta }{\sup }{\left({\int }_{I_h}{\left|x(t+h)-x(t)\right|}^p\mathrm{d}t\right)}^{1/p},\delta \in [0,1],$$

$$B_{p,\alpha }^o:=\left\{x\in L_p([0,1]):\underset{\delta \to 0}{\lim }{\delta }^{-\alpha }{\omega }_p(x,\delta )=0\right\}.$$

$${\Vert x\Vert }_{p,\alpha }={\Vert x\Vert }_p+\underset{\delta \in (0,1)}{\sup }{\delta }^{-\alpha }{\omega }_p(x,\delta ),x\in B_{p,\alpha }^o,$$

$$B_{p,\alpha }^o\hookrightarrow B_{p,\beta }^o,\text{for}0\le \beta <\alpha ;$$

$$B_{p,\alpha }^o\hookrightarrow B_{q,\alpha }^o,\text{for}1\le q<p<\infty .$$

$$q(p,\alpha ):=1/(1/2-\alpha +1/p).$$

$${\xi }_n(t)=\sum _{k=1}^{\lfloor nt\rfloor }{X}_k+(nt-\lfloor nt\rfloor )X_{\lfloor nt\rfloor +1}.$$

$$n^{-1/2}{\sigma }^{-1}{\xi }_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}W\text{in the space}{B}_{p,\alpha }^o$$

$$\underset{t\to \infty }{\lim }{t}^{q(p,\alpha )}\mathsf{P}\left\{\left|Y\right|⩾t\right\}=0.$$

$$n^{-1/2}{\sigma }^{-1}{\xi }_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}W\text{in the space}{B}_{p,\alpha }^o.$$

$$\mathcal{D}_{0}=\{0,1\},\qquad\mathcal{D}_{j}=\bigl{\{}(2l-1)2^{-j};\;1\leqslant l\leqslant 2^{j-1}\bigr{\}},\quad j\geqslant 1.$$

$$\mathcal{D}=\bigcup _{j⩾0}{\mathcal{D}}_j$$

$$r^{-}:=r-2^{-j},r^{+}:=r+2^{-j}.$$

$$\Lambda_{r}(t)=\left\{\begin{array}[]{ll}2^{j}(t-r^{-})&\text{if $t\in(r^{-},r]$;}\\ 2^{j}(r^{+}-t)&\text{if $t\in(r,r^{+}]$;}\\ 0&\text{else.}\end{array}\right.$$

$${\Lambda }_0(t)=1-t,{\Lambda }_1(t)=t,t\in [0,1].$$

$$x=\sum _{r\in \mathcal{D}}{\lambda }_r(x){\Lambda }_r,$$

$${\lambda }_r(x):=x(r)-\frac{x(r^{+})+x(r^{-})}{2},r\in D_j,j⩾1$$

$${\lambda }_0(x):=x(0),{\lambda }_1(x):=x(1).$$

$$\left\lVert x\right\rVert_{p,\alpha}\sim\left\lVert x\right\rVert_{p,\alpha}^{\mathrm{seq}}:=\sup_{j\geqslant 0}2^{j\alpha-i/p}\Big{(}\sum_{r\in D_{j}}|\lambda_{r}(x)|^{p}\Big{)}^{1/p}.$$

$$f_k(t)=\sum _{i=0}^k{c}_{ik}{\Lambda }_i(t),t\in [0,1],$$

$$x=\sum _{k=0}^{\infty }{x}_k{f}_k,$$

$${\omega }_p(x,\delta )⩽\underset{n^{\prime }}{\mathrm{liminf}}{\omega }_p(x_{n^{\prime }},\delta )⩽\underset{n}{\sup }{\omega }_p(x_n,\delta ).$$

$$||x_n-x_m|{|}_{p,\alpha }$$

$$⩽||x_n-x_m|{|}_p+\epsilon +2{\delta }_{\epsilon }^{-\alpha }||x_n-x_m|{|}_p$$

$$\lim_{\delta\to 0}\sup_{n\geqslant 1}\mathsf{P}\Big{(}\omega_{p,\alpha}(Z_{n},\delta)\geqslant\varepsilon\Big{)}=0.$$

$$W_{f,n}=n^{-1/2}{\zeta }_n.$$

$$\lim_{J\to+\infty}\limsup_{N\to+\infty}\sum_{j=J}^{N}2^{j}\int_{0}^{1}x^{p-1}\mathsf{P}\Big{(}2^{-(N-j)/2}\max_{1\leqslant k\leqslant 2^{N-j}}\left|S_{f,k}\right|>x2^{{j}/{q(p,\alpha)}}\Big{)}\mathrm{d}x=0.$$

$$||W_{f,n}|{|}_p\le \underset{0\le t\le 1}{\sup }|W_{f,n}(t)|=n^{-1/2}\underset{0⩽t⩽1}{\max }|S_{f,k}|,$$

Full text

Keywords: invariance principle, martingale differences, stationary sequences, Besov spaces.

AMS MSC 2010: 60F17; 60G10; 60G42.

1. Introduction and main results

By weak invariance principle in a topological function space, say, $E$ we understand the weak convergence of a sequence of probability measures induced on $E$ by normalized polygonal line processes build from partial sums of random variables. The choice of the space $E$ is important due to possible statistical applications via continuous mappings. Since stronger topology generates more continuous functionals, it is beneficial to have the weak invariance principle proved in as strong as possible topological framework.

Classical Donsker’s weak invariance principle considers the space $E=C[0,1]$ and polygonal line processes build from partial sums of i.i.d. centred random variables with finite second moment. An intensive research has been done in order to extend Donsker’s result to a stronger topological framework as well to a larger class of random variables (see, e.g., [9], [12], [4] and references therein).

In this paper we consider weak invariance principle in Besov spaces for a class of strictly stationary sequence of martingale differences. To be more precise, let us first introduce some notation and definitions used throughout the paper.

Let $\left(\Omega,\mathcal{F},\mathsf{P}\right)$ be a probability space and $\mathsf{T}\colon\Omega\to\Omega$ be a bijective bi-measurable transformation preserving the probability $\mathsf{P}$. The quadruple $(\Omega,\mathcal{F},\mathsf{P},\mathsf{T})$ is referred to as dynamical system (see, e.g., [10] for some background material). We assume that there is a sub-$\sigma$-algebra $\mathcal{F}_{0}\subset\mathcal{F}$ such that $\mathsf{T}\mathcal{F}_{0}\subset\mathcal{F}_{0}$ and by $\mathcal{I}$ we denote the $\sigma$-algebra of the sets $A\in\mathcal{F}$ such that $\mathsf{T}^{-1}A=A$.

Next we consider a strictly stationary sequence $\left(X_{j},\ j\geqslant 0\right)$ constructed as $X_{j}:=f\circ\mathsf{T}^{j}$, where $f:\Omega\to\mathbb{R}$ is $\mathcal{F}_{0}$-measurable. We define also a non-decreasing filtration $\mathcal{F}_{k}=\mathsf{T}^{-k}\mathcal{F}_{0}$, $k\geqslant 1$. Note that $\left(X_{j},\mathcal{F}_{j},j\geqslant 0\right)$ is then a martingale differences sequence provided $\mathsf{E}\left(f\mid\mathsf{T}\mathcal{\mathcal{F}}_{0}\right)=0$.

Set

[TABLE]

Our main object of investigation is the sequence of polygonal line processes $\zeta_{f,n}:=(\zeta_{f,n}(t),t\in[0,1]),$ $n\geqslant 1$, defined by

[TABLE]

where for a real number $a\geqslant 0$, $\lfloor a\rfloor:=\max\{k\colon k\in\{0,1,\ldots\},k\leq a\}$. To define the paths space under consideration let $L_{p}([0,1])$ be the space of Lebesgue integrable functions with exponent $p$ ($1\leqslant p<\infty$) and the norm

[TABLE]

If $x\in L_{p}([0,1])$ its $L_{p}$-modulus of smoothness is defined as

[TABLE]

where $I_{h}=[0,1]\cap[-h,1-h]$. Let $\alpha\in[0,1)$. The Besov space $B^{o}_{p,\alpha}=B^{o}_{p,\alpha}([0,1])$ is defined by

[TABLE]

Endowed with the norm

[TABLE]

the space $B^{o}_{p,\alpha}$ is a separable Banach space and the following embeddings are continuous:

[TABLE]

Each $B^{o}_{p,\alpha}(0,1)$ where $p\geqslant 1,0\leqslant\alpha<1/2,$ supports a standard Wiener process $W=\left(W(t),0\leqslant t\leqslant 1\right)$ (see, e.g., [14]). We note also, that any polygonal line process belongs to each of $B^{o}_{p,\alpha}$, $p\geq 1$, $\alpha\in[0,1)$.

As usually $\xrightarrow[]{\mathcal{D}}$ denotes convergence in distribution.

Theorem 1.1**.**

Let $p>2$, $1/p<\alpha<1/2$ and

[TABLE]

Let $(f\circ\mathsf{T}^{i},\ \mathsf{T}^{-i}\mathcal{\mathcal{F}}_{0},\ i\geqslant 0)$ be a martingale differences sequence. Assume that the following two conditions hold :

• $(i)$

$\lim_{t\to\infty}t^{q(p,\alpha)}\mathsf{P}\left\{\left|f\right|\geqslant t\right\}=0;$**

• $(ii)$

\mathsf{E}\left(\big{[}\mathsf{E}\left(f^{2}\mid\mathsf{T}\mathcal{F}_{0}\right)\big{]}^{q(p,\alpha)/2}\right)<\infty.**

Then the convergence $n^{-1/2}\zeta_{f,n}\xrightarrow[n\to\infty]{\mathcal{D}}\sqrt{\mathsf{E}\left(f^{2}\mid\mathcal{I}\right)}\ W$ holds in the space $B_{p,\alpha}^{o},$ where $W$ is independent of $\mathsf{E}\left(f^{2}\mid\mathcal{I}\right)$.

Let us note that condition (i) is stronger for the function $f$ than its square integrability since (i) yields $\mathsf{E}\left(|f|^{r}\right)<\infty$ for any $r<q(p,\alpha)$ and $q(p,\alpha)>2$ when $\alpha>1/p$. However as shows our next result, condition (i) is necessary and sufficient for independent identically distributed (i.i.d.) random variables. To formulate the result let $Y,Y_{1},Y_{2},\dots$ be mean zero i.i.d.  random variables with finite variance $\sigma^{2}=\mathsf{E}\left(Y^{2}\right)>0$. Let $\xi_{n}=(\xi_{n}(t),t\in[0,1])$, be defined by

[TABLE]

Theorem 1.2**.**

Let $p>2,1/p<\alpha<1/2$ and let $q(p,\alpha)$ be defined by (1.0.1). Then

[TABLE]

if and only if

[TABLE]

Since $B_{\infty,\alpha}^{o}$ matches Hölder spaces we see, that Theorems 1.2 and 1.1 complement the weak invariance principle obtained in a Hölderian framework by [12] and [4]. Concerning condition (ii) of Theorem 1.1 we prove a need of certain extra assumption by a counterexample which for any dynamical system with positive entropy constructs a function $f$ that satisfies the condition (i) but the convergence of polygonal line processes fails. Precise result reads as follows.

Theorem 1.3**.**

Let $p>2$, $1/p<\alpha<1/2$ and $q(p,\alpha)$ be given by (1.0.1). For each dynamical system $\left(\Omega,\mathcal{F},\mathsf{P},\mathsf{T}\right)$ with positive entropy, there exists a function $m\colon\Omega\to\mathbb{R}$ and a $\sigma$-algebra $\mathcal{F}_{0}$ for which $\mathsf{T}\mathcal{F}_{0}\subset\mathcal{F}_{0}$ such that:

• $(i)$

*the sequence $\left(m\circ\mathsf{T}^{i},\ \mathsf{T}^{-i}\mathcal{F}_{0},i\geqslant 0\right)$ is a martingale difference sequence; *

• $(ii)$

the convergence \lim_{t\to+\infty}t^{q(p,\alpha)}\mathsf{P}\Big{(}\left|m\right|\geqslant t\Big{)}=0 takes place;

• $(iii)$

the sequence $\left(n^{-1/2}\zeta_{m,n}\right)$ is not tight in $B^{o}_{p,\alpha}$.

As it is seen from our next results the case where $0\leq\alpha\leq 1/p$ is indeed quite different from the previously considered case where $1/p<\alpha<1/2$.

Theorem 1.4**.**

Let $p\geqslant 1$ and $\alpha\in[0,1/2)\cap[0,1/p]$. Let $\left(f\circ\mathsf{T}^{i},\ \mathsf{T}^{-i}\mathcal{F}_{0},i\geqslant 0\right)$ be a martingale differences sequence. If $\mathsf{E}\,{f^{2}}<\infty$ then $n^{-1/2}\zeta_{f,n}\xrightarrow[n\to\infty]{\mathcal{D}}\sqrt{\mathsf{E}\left(f^{2}|\mathcal{I}\right)}W$ in the space $B_{p,\alpha}^{o}$, where $W$ is independent of $\mathsf{E}\left(f^{2}\mid\mathcal{I}\right)$.

Theorem 1.5**.**

*Let $\alpha$ and $p$ be as in Theorem 1.4. Then *

[TABLE]

Let us note that the finiteness of the second moment $\mathsf{E}\,Y^{2}$ is necessary for the convergence (1.0.4).

The rest of the paper is organized as follows. In Section 2 we shortly present needed information on structure of Besov spaces and tightness of measures on these spaces. Section 3 contains proofs of Theorems 1.1, 1.2 and 1.3 whereas Section 4 is devoted to the proofs of Theorems 1.4 and 1.5. Finally, in Section 6 we discuss possible applications of invariance principle in the Besov framework.

2. Some functional analysis and probabilistic tools

We denote by $\mathcal{D}_{j}$ the set of dyadic numbers in $[0,1]$ of level $j$, i.e.

[TABLE]

Set

[TABLE]

and write for $r\in\mathcal{D}_{j}$,

[TABLE]

The triangular Faber-Schauder functions $\Lambda_{r}$ for $r\in\mathcal{D}_{j}$, $j>0$, are

[TABLE]

When $j=0$, we just take the restriction to $[0,1]$ in the above formula, so

[TABLE]

Theorem 2.1** ([3]).**

Let $p>1$ and $1/p<\alpha<1.$ The Faber-Schauder system $\{\Lambda_{r},r\in D\}$ is the Schauder basis for $B^{o}_{p,\alpha}$: each $x\in B^{0}_{p,\alpha}$ has the unique representation,

[TABLE]

where

[TABLE]

and in the special case $j=0$,

[TABLE]

Moreover the norm is equivalent to the sequential norm:

[TABLE]

The Schmidt orthogonalization procedure (with respect to inner product in $L_{2}(0,1)$) applied to Faber-Schauder system leads to the Franklin system $\{f_{k},k\geqslant 0\}$:

[TABLE]

with $c_{kk}>0$ for $k\geqslant 0$, where the matrix $(c_{ik})$ is uniquely determined.

Theorem 2.2** ([3]).**

The Franklin system $\{f_{n},n\geqslant 0\}$ is the basis for $B^{o}_{p,\alpha}$, $p\geqslant 1,0\leqslant\alpha<1$: each $x\in B^{o}_{p,\alpha}$ has the unique representation,

[TABLE]

where $x_{k}=\langle x,f_{k}\rangle:=\int_{0}^{1}x(t)f_{k}(t)\mathrm{d}\,t$, $k\geqslant 0$.

The following proposition is proved in [2] for $\alpha>1/p$ but similar arguments works as well for any $0\leqslant\alpha<1$.

Proposition 2.3**.**

Let $p\geqslant 1$ and $0\leqslant\alpha<1$. The set $K\subset B_{p,\alpha}^{o}$ is relatively compact if and only if

• $(i)$

$\sup_{x\in K}||x||_{p}<\infty,$**

• $(ii)$

$\lim_{\delta\to 0}\sup_{x\in K}\delta^{-\alpha}\omega_{p,\alpha}(x,\delta)=0.$**

Proof.

One easily checks that (i) and (ii) yields relative compactness of $K$ in $L_{p}([0,1])$. Therefore for any sequence $(x_{n})_{n\geqslant 1}$ of $K$ there exists a subsequence, which we denote also $(x_{n})$, converging in $L_{p}([0,1])$ to some $x\in L_{p}([0,1])$. To finish the proof it suffices to prove that

• $(a)$

$x\in B^{o}_{p,\alpha}$;

• $(b)$

$(x_{n})_{n\geqslant 1}$ is a Cauchy sequence in $B^{o}_{p,\alpha}$.

Taking a.s. convergence subsequence $(x_{n^{\prime}})$ and applying Fatou lemma we easily obtain for any $0<\delta\leqslant 1$,

[TABLE]

This yields (a). To prove (b) observe that for each $\varepsilon>0$ there exists $\delta_{\varepsilon}>0$ such that $\delta^{-\alpha}\sup_{n}\omega_{p}(x_{n},\delta)<\varepsilon\quad\textrm{when}\quad\delta<\delta_{\varepsilon}$, hence, for $n,m\geqslant 1$

[TABLE]

and we complete the proof since $\lim_{n,m\to\infty}||x_{n}-x_{m}||_{p}=0$. ∎

Consider stochastic processes $Z,(Z_{n})_{n\geqslant 1}$ with paths space $B^{o}_{p,\alpha}$ which is endowed with Borel $\sigma$-algebra $\mathcal{B}(B^{o}_{p,\alpha})$. Let $P_{Z},P_{Z_{n}},n\geqslant 1,$ be the corresponding distributions. As generally accepted the sequence $(Z_{n})$ converges in distribution to $Z$ in $B^{o}_{p,\alpha}$ (denoted $Z_{n}\xrightarrow[n\to\infty]{\mathcal{D}}Z$ in $B^{o}_{p,\alpha}$) provided $(P_{Z_{n}})$ converges weakly to $P_{Z}$: $\lim_{n\to\infty}\int f\mathrm{d}\,P_{Z_{n}}=\int f\mathrm{d}\,P_{Z}$ for each bounded continuous $f:B^{o}_{p,\alpha}\to\mathbb{R}$. The sequence $(Z_{n})$ is tight in $B^{o}_{p,\alpha}$ if for each $\varepsilon>0$ there is a relatively compact set $K_{\varepsilon}\subset B^{o}_{p,\alpha}$ such that $\inf_{n\geqslant 1}\mathsf{P}(Z_{n}\in K_{\varepsilon})>1-\varepsilon$. Due to the well known Prohorov’s theorem convergence in distribution in a separable metric space is coherent with tightness. Indeed, to prove convergence in distribution one has to establish tightness and to ensure uniqueness of the limiting distributions.

The following tightness criterion is obtained from Proposition 2.3.

Theorem 2.4**.**

The sequence $\left(Z_{n}\right)$ of random processes with paths in $B^{o}_{p,\alpha}(0,1)$ is tight if and only if the following two conditions are satisfied:

• (i)

\lim_{b\to\infty}\sup_{n\geqslant 1}\mathsf{P}\Big{(}||Z_{n}||_{p}>b\Big{)}=0;

• (ii)

for each $\varepsilon>0$

[TABLE]

Proof.

See, e.g., the proof of Theorem 8.2. in [1]. ∎

Theorem 2.5**.**

Let $1>\alpha>1/p$. The sequence $(Z_{n})$ of random elements in the Besov space $B^{o}_{p,\alpha}(0,1)$ is tight if and only if the following two conditions are satisfied:

• (i)

\displaystyle\lim_{b\to\infty}\sup_{n}\mathsf{P}\bigl{\{}||Z_{n}||_{p}>b\bigr{\}}=0;

• (ii)

for each $\varepsilon>0$, \displaystyle\lim_{J\to\infty}\limsup_{n\to\infty}\mathsf{P}\Big{(}\sup_{j\geqslant J}2^{j\alpha-j/p}\Big{(}\sum_{r\in D_{j}}\left|\lambda_{r}(\zeta_{n})\right|^{p}\Big{)}^{1/p}>\varepsilon\Big{)}=0.

Proof.

It is just a corollary of tightness criterion established in [15] for Schauder decomposable Banach spaces as Besov spaces $B^{o}_{p,\alpha}$ with $\alpha>1/p,$ are such. ∎

3. Proofs: the case $\alpha>1/p$

We start this section with some auxiliary results which could be helpful when dealing with weak invariance principle for stationary sequences. Throughout we denote

[TABLE]

3.1. Auxiliary results

Lemma 3.1**.**

Let $p\geqslant 1$ and $\alpha>1/p$. Assume that $Z$ is a random element in boths spaces $C[0,1]$ and $B^{o}_{p,\alpha}$. Then for any stationary sequence $\left(f\circ\mathsf{T}^{j}\right)$ if

• $(i)$

$W_{f,n}\xrightarrow[n\to\infty]{\mathcal{D}}Z$* in $C[0,1]$, and*

• $(ii)$

$(W_{f,n})$* is tight in $B^{o}_{p,\alpha}$,*

then $W_{f,n}\xrightarrow[n\to\infty]{\mathcal{D}}Z$ in $B^{o}_{p,\alpha}$.

Proof.

From (ii) we have that each subsequence of $(W_{f,n})$ has further subsequence that converges in distribution. If $W_{f,n^{\prime}}\xrightarrow[n\to\infty]{\mathcal{D}}Y^{\prime}$ and $W_{f,n^{\prime\prime}}\xrightarrow[n\to\infty]{\mathcal{D}}Y^{\prime\prime}$ then we have that $\lambda_{r}(W_{f,n^{\prime}})\xrightarrow[n\to\infty]{\mathcal{D}}\lambda_{r}(Y^{\prime})$ and $\lambda_{r}(W_{f,n^{\prime\prime}})\xrightarrow[n\to\infty]{\mathcal{D}}\lambda_{r}(Y^{\prime\prime})$ for any dyadic number $r$. But (i) gives that both $\lambda_{r}(Y^{\prime})$ and $\lambda_{r}(Y^{\prime\prime})$ have the same distribution as $\lambda_{r}(Z)$. Since Schauder coefficients $(\lambda_{r}(Z))$ determines the distribution of $Z$ we can conclude that $Y^{\prime}$ and $Y^{\prime\prime}$ are equally distributed with $Z$. This ends the proof. ∎

For polygonal line processes build from any stationary sequence the tightness conditions given in Theorem 2.5 can be simplified.

Theorem 3.2**.**

Let $p\geqslant 1$ and $\alpha>1/p$. The sequence $(W_{f,n})$ is tight in $B_{p,\alpha}^{o}$ provided that

[TABLE]

Proof.

Assume that $f$ satisfies (3.1.1). We have to show that $(W_{f,n})$ satisfies the conditions of Theorem 2.5. First we check its condition (i). Since

[TABLE]

the proof of (i) reduces to

[TABLE]

Notice that (3.1.1) implies (by considering the term of index $J$ in the sum) that

[TABLE]

and consequently

[TABLE]

For a fixed $\varepsilon$, we choose $J_{0}$ such that

[TABLE]

There exists an integer $N_{0}$ such that for $N\geqslant N_{0}$,

[TABLE]

Since $\sup_{N\leqslant N_{0}}\mathsf{P}\left(2^{-N/2}\max_{1\leqslant k\leqslant 2^{N}}\left|S_{f,k}\right|>b\right)\to 0$ as $b$ goes to infinity, we can choose $b^{\prime}_{0}$ such that $\max_{N\leqslant N_{0}}\mathsf{P}\left(2^{-N/2}\max_{1\leqslant k\leqslant 2^{N}}\left|S_{f,k}\right|>b^{\prime}_{0}\right)<\varepsilon$. Taking $b_{0}:=\max\left(2^{{J_{0}}/{q(p,\alpha)}}/2,b^{\prime}_{0}\right)$, we have for $b\geqslant b_{0}$,

[TABLE]

which proves (3.1.2) and the same time (i) of Theorem 2.5.

Now, let us prove condition (ii) of Theorem 2.5. Since

[TABLE]

we infer that condition (3.1.1) implies

[TABLE]

We first prove that for each positive $\varepsilon$,

[TABLE]

We shall actually prove that

[TABLE]

since the differences $W_{f,n}(r)-W_{f,n}(r^{-})$ can be treated similarly. To this aim, define for a fixed $j\geqslant\lfloor\log n\rfloor+1$ the sets

[TABLE]

and

[TABLE]

Assume that $r$ belongs to $I_{k}$. Then $\lfloor nr\rfloor=\lfloor nr^{+}\rfloor=k$. We thus have

[TABLE]

Now, assume that $r$ belongs to $J_{k}$. Then

[TABLE]

and using the fact that $0\leqslant nr^{+}-(k+1)\leqslant nr^{+}-nr=n2^{j}$ and $0\leqslant 1-\left(nr-k\right)\leqslant\left(nr^{+}-k\right)-\left(nr-k\right)=n2^{-j}$, we get

[TABLE]

Since $D_{j}=\{1-2^{-j}\}\cup\bigcup_{k=0}^{n-1}I_{k}\cup\bigcup_{k=0}^{n-2}J_{k}$ and for $r=1-2^{-j}$,

[TABLE]

we have in view of (3.1.5) and (3.1.6),

[TABLE]

We now have to bound $\operatorname{Card}\left(I_{k}\right)$ and $\operatorname{Card}\left(J_{k}\right)$. Let $1\leqslant l\leqslant 2^{j}$. If $(2l-1)2^{-j}$ belongs to $I_{k}$, then we should have $2^{j}k/n\leqslant 2l-1<2l<2^{j}(k+1)/n$ hence $2^{j-1}k/n\leqslant l<2^{j-1}(k+1)/n$ and it follows that $I_{k}$ cannot have more than $2^{j}/n$ elements. If $(2l-1)2^{-j}$ belongs to $J_{k}$, then we should have $2^{j}(k+1)/n\leqslant 2l<2^{j}(k+2)/n$ and we deduce that the cardinal of $J_{k}$ does not exceed $2^{j}/n$. Therefore, we have

[TABLE]

and

[TABLE]

We thus have to prove that the latter term goes to zero in probability as $n$ goes to infinity.

Lemma 3.3**.**

Let $f$ be a function such that (3.1.3) holds. Then

[TABLE]

Proof.

For fixed $\delta$ and $n$, define f^{\prime}:=f\mathbf{1}\big{(}\left|f\right|\leqslant\delta n^{1/q(p,\alpha)}\big{)} and $f^{\prime\prime}=f-f^{\prime}$. By Markov’s inequality, we have with $q=q(p,\alpha)$

[TABLE]

Now, note that

[TABLE]

hence by (3.1), we have

[TABLE]

Notice also that

[TABLE]

The combination of (3.1.8) and (3.1.9) gives

[TABLE]

and since $\delta$ is arbitrary and $p>q$, this concludes the proof of Lemma 3.3. ∎

An application of the Lemma 3.3 gives (3.1.4). Now, we have to prove that

[TABLE]

It suffices to prove that

[TABLE]

Indeed, we have

[TABLE]

and for $j\leqslant\lfloor\log n\rfloor$, $2^{j}\leqslant n$, so that the set $\{\lfloor nr^{+}\rfloor,\lfloor nr\rfloor,r\in D_{j}\}$ consists of distinct elements. Therefore,

[TABLE]

and this quantity goes to zero in probability by Lemma 3.3. The proof of (3.1.10) reduces to establish that for each positive $\varepsilon$,

[TABLE]

where

[TABLE]

We now bound $\mathsf{P}\left(A_{n,j}\right)$ by splitting the probability over the set

[TABLE]

One bounds $\mathsf{P}\left(A_{n,j}\cap B_{n,j}\right)$ by $\mathsf{P}(B_{n,j})$, which can in turn be bounded by

[TABLE]

and thanks to stationarity and the fact that

[TABLE]

we obtain

[TABLE]

Now, in order to bound $\mathsf{P}\left(A_{n,j}\cap B_{n,j}^{c}\right)$, we start by the pointwise inequalities

[TABLE]

Integrating and using the fact that for a non-negative random variable $Y$ and a positive $R$,

[TABLE]

we derive by stationarity and (3.1.12) that

[TABLE]

Let us denote by $K$ a constant depending only on $p$ and $\varepsilon$ which may change from line to line. By (3.1.13) and (3.1.14), we derive that

[TABLE]

If $2^{N}\leqslant n<2^{N+1}$, then we have

[TABLE]

hence

[TABLE]

Splitting the integral into two parts, we infer that

[TABLE]

and the limit of the latter quantity as $J$ goes to infinity is zero by (3.1.1). This concludes the proof of Theorem 3.2. ∎

Remark 3.4*.*

Using deviation inequalities, similar results as those found for the Hölderian weak invariance principle for stationary mixing and $\tau$-dependent sequences in [4] can be found for Besov spaces.

Lemma 3.5** (Proposition 3.5 in [5] ).**

For any $q>2$, there exists a constant $c(q)$ such that if $\left(f\circ\mathsf{T}^{i}\right)_{i\geqslant 0}$ is a martingale differences sequence with respect to the filtration $\left(\mathsf{T}^{-i}\mathcal{F}_{0}\right)_{i\geqslant 0}$ then for each integer $n\geqslant 1$,

[TABLE]

3.2. Proof of Theorem 1.1

Acording to Lemma 3.1 we need only to prove that the sequence $\left(n^{-1/2}\zeta_{f,n}\right)_{n\geqslant 1}$ is tight in $B_{p,\alpha}^{o}$. To this aim, we have to check the condition (3.1.1) of Theorem 3.2. For fixed $N$ and $J$ such that $N\geqslant J$, j\in\Big{(}J,\dots,N\Big{)} and $x\in[0,1]$, we have by (3.1.16) of Lemma 3.5,

[TABLE]

from which we infer that

[TABLE]

Using the fact that

[TABLE]

we derive the bound

[TABLE]

Since $j\leqslant N$, we have $2^{\frac{N}{2}}xu2^{j\left(1/p-\alpha\right)}\geqslant xu2^{N/q(p,\alpha)}$ and accounting the inequality $\sum_{j=0}^{N}2^{j\left(1/p-\alpha\right)}\leqslant 2^{N\left(1/p-\alpha\right)}/(1-2^{1/p-\alpha})$, we obtain

[TABLE]

Since $p>q(p,\alpha)$ and $q>q(p,\alpha)$, the integral $\int_{0}^{1}\int_{0}^{1}u^{q-q(p,\alpha)-1}x^{p-q(p,\alpha)-1}\mathrm{d}x\mathrm{d}u$ is convergent and we infer by the monotone convergence theorem that

[TABLE]

Now, in order to control $B(N,J)$, we use the following elementary inequality: if $Y$ is a non-negative random variable, then for each $J\geqslant 1$,

[TABLE]

Applying this to $Y:=\left(\mathsf{E}\left(f^{2}\mid\mathsf{T}\mathcal{F}_{0}\right)\right)^{1/2}/(vx)$, we obtain that

[TABLE]

Here again, we conclude by monotone convergence that

[TABLE]

since the integrals $\int_{0}^{1}x^{p-q(\alpha)-1}\mathrm{d}x$ and \int_{0}^{+\infty}\min\Big{(}v,v^{q-1}\Big{)}v^{-q(p,\alpha)}\mathrm{d}x are finite (as $q>q(\alpha)$).

Tightness of the sequence $\left(W_{f,n}\right)_{n\geqslant 1}$ now follows from Theorem 3.2 and the combination of (3.2.2), (3.2.3) and (3.2.4). Acounting Lemma 3.1 this concludes the proof of Theorem 1.1.

3.3. Proof of Theorem 1.2

Sufficiency of the condition is contained in Theorem 1.1. Indeed, we represent the sequence $\left(Y_{j}\right)_{j\geqslant 0}$ by $\left(f\circ\mathsf{T}^{j}\right)_{j\in\mathbb{Z}}$, that is, $\left(f\circ\mathsf{T}^{j}\right)_{j\in\mathbb{Z}}$ is an i.i.d.  sequence and $Y_{j}$ has the same distribution as $f\circ\mathsf{T}^{j}$. To this aim we define $\Omega=\mathbb{R}^{\mathbb{Z}},\mathcal{F}=\mathcal{B}^{\mathbb{Z}}$ and $\mathsf{P}=\mathsf{P}_{Y}^{\mathbb{Z}}$, where $\mathsf{P}_{Y}$ is the distribution of $Y_{0}$. Let $f((\omega_{j}))=\omega_{0}$ for $(\omega_{j})\in\mathbb{R}^{Z}$ and let $\mathsf{T}:\Omega\to\Omega$ be the shift operator: $\mathsf{T}((\omega_{j}))=(\omega_{j+1})$. Next let $\mathcal{F}_{0}:=\sigma\left(f\circ\mathsf{T}^{j},j\leqslant 0\right)$. Then $\mathsf{T}\mathcal{F}_{0}\subset\mathcal{F}_{0}$ and $\mathsf{E}\left(f\mid\mathsf{T}\mathcal{F}_{0}\right)=0$, since $f$ is independent of $\mathsf{T}\mathcal{F}_{0}$ and centered. Moreover, $\mathsf{E}\left(f^{2}\mid\mathsf{T}\mathcal{F}_{0}\right)=\mathsf{E}\left(f^{2}\right)$, again by independence. Therefore, condition (ii) of (1.1) is satisfied. Since $\mathcal{I}$ is trivial, $\mathsf{E}\left(f^{2}\mid\mathcal{I}\right)=\mathsf{E}\left(f^{2}\right)$, which gives the convergence (1.0.2).

Let us prove the necessity of (1.0.3) for the invariance principle in $B_{p,\alpha}^{o}$. Since the space $B_{p,\alpha}^{o}$ is a separable Banach space, the sequence $\left(W_{2^{n}}\right)_{n\geqslant 1}$ is tight in $B_{p,\alpha}^{o}$. Using Theorem 1 in [13], we can find for any positive $\eta$ a number $J_{0}$ such that

[TABLE]

Therefore, if $n$ is large enough, we have

[TABLE]

Since

[TABLE]

we have the convergence in probability of the sequence $\left(2^{n\alpha-j/p}2^{-np/2}\sum_{l=1}^{2^{n-1}}\left|X_{2l-1}\right|^{p}\right)$ to [math]. By [7], this implies that 2^{n\alpha-j/p}2^{-np/2}\mathsf{P}\Big{(}\left|X_{1}\right|>2^{n}\Big{)}\to 0, hence (1.0.3) holds. This ends the proof of Theorem 1.2.

3.4. Proof of Theorem 1.3

We first start by a lemma which guarantees the lake of tightness of the partial sum process.

Lemma 3.6**.**

Let $1/p<\alpha<1/2$ and let $f$ be a function such that there exist increasing sequences of real numbers $(n_{l})_{l\geqslant 1}$ and $(k_{l})_{l\geqslant 1}$ satisfying the following properties: $k_{l}/n_{l}\to 0$ as $l$ goes to infinity and

[TABLE]

where $q(p,\alpha)$ is given by (1.0.1). Then the sequence $(W_{n}(f))_{n\geqslant 1}$ is not tight in $B_{p,\alpha}^{o}$.

Proof.

If the sequence $\left(W_{n}(f)\right)_{n\geqslant 1}$ was tight in $B_{p,\alpha}^{o}$, then we would be able to extract a weakly convergence subsequence of $\left(W_{n_{l}}(f)\right)_{l\geqslant 1}$. Therefore, we can assume without loss of generality that $\left(W_{n_{l}}(f)\right)_{l\geqslant 1}$ converges in distribution in $B_{p,\alpha}$. Consequently, the sequence $\left(\sup_{\left|t\right|\leqslant k_{l}/n_{l}}t^{-\alpha}\omega_{p}\left(W_{n_{l}},t\right)\right)_{l\geqslant 1}$ should convergence to [math] in probability as $l$ goes to infinity. But

[TABLE]

for some constant depending only on $p$ (this can be seen by restricting the supremum over the $t$ of the form $k/n_{l}$ where $1\leqslant k\leqslant k_{l}$).

∎

Let us recall the statement of Lemma 3.8 in [8].

Lemma 3.7**.**

Let $(\Omega,\mathcal{F},\mathsf{P},\mathsf{T})$ be an ergodic probability measure preserving system of positive entropy. There exists two $\mathsf{T}$-invariant sub-$\sigma$-algebras $\mathcal{B}$ and $\mathcal{C}$ of $\mathcal{F}$ and a function $g\colon\Omega\to\mathbb{R}$ such that:

• •

the $\sigma$-algebras $\mathcal{B}$ and $\mathcal{C}$ are independent;

• •

the function $g$ is $\mathcal{B}$-measurable, takes the values $-1$, [math] and $1$, has zero mean and the process $(g\circ\mathsf{T}^{n})_{n\in\mathbb{Z}}$ is independent;

• •

the dynamical system $(\Omega,\mathcal{C},\mathsf{P},\mathsf{T})$ is aperiodic.

In the sequel, we shall assume for simplicity that \mathsf{P}\Big{(}g=1\Big{)}=\mathsf{P}\Big{(}g=-1\Big{)}=1/2.

The construction follows the lines of that of Theorem 2.1 in [6]. We define three increasing sequences of positive integers $\left(I_{l}\right)_{l\geqslant 1}$, $\left(J_{l}\right)_{l\geqslant 1}$, $\left(N_{l}\right)_{l\geqslant 1}$ and a sequence of real numbers $\left(L_{l}\right)_{l\geqslant 1}$ such that

[TABLE]

$L_{l}$ is a continuity point of the cumulative distribution function of the random variable $2^{-1}Y_{1/2,1}$, which is defined in (5.0.1). Now, we define a sequence of real numbers $\left(J_{l}\right)_{l\geqslant 1}$ in such a way that for each $l\geqslant 1$,

[TABLE]

Now, by Proposition 5.1, we can choose for each $l\geqslant 1$ an integer $I_{l}$ such that

[TABLE]

Let $K_{l}:=2^{I_{l}+J_{l}}$. We define the sequence $\left(N_{l}\right)_{l\geqslant 1}$ in such a way that for each $l\geqslant 1$,

[TABLE]

[TABLE]

Using Rokhlin’s lemma, we can find for any integer $l\geqslant 1$ a measurable set $C_{l}\in\mathcal{C}$ such that the sets $\mathsf{T}^{i}C_{l}$, $i=0,\dots,N_{l}-1$ are pairwise disjoint and $\mathsf{P}\left(\bigcup_{i=0}^{N_{l}-1}\mathsf{T}^{i}C_{l}\right)>1/2$. We define for $l\geqslant 1$

[TABLE]

[TABLE]

Note that $\mathsf{P}\left(f_{l}\neq 0\right)\leqslant K_{l}/N_{l}$, hence by (3.4.4) and the Borel-Cantelli lemma, the function $f$ is well defined almost everywhere. Define

[TABLE]

Proposition 3.8**.**

The $\sigma$-algebra $\mathcal{F}_{0}$ satisfies $\mathsf{T}\mathcal{F}_{0}\subset\mathcal{F}_{0}$. The function $m$ defined by (3.4.5) and (3.4.6) is $\mathcal{F}_{0}$-measurable and satisfies $\mathsf{E}\,\left[m\mid\mathsf{T}\mathcal{F}_{0}\right]=0$ and $\lim_{t\to+\infty}t^{q(p,\alpha)}\mathsf{P}\left\{\left|m\right|>t\right\}=0$.

A proof can be found in [6]

It remains to prove that the sequence $(W_{n}(m))_{n\geqslant 1}$ is not tight in $B_{p,\alpha}^{o}$.

To this aim, we shall check the conditions of Lemma 3.6. We first show the following intermediate step.

Lemma 3.9**.**

For each integer $l\geqslant 1$,

[TABLE]

Let $l\geqslant 1$ be fixed. Assume that $\omega$ belongs to $\mathsf{T}^{N_{l}-i_{0}}$ for some $i_{0}\in\left\{K_{l},\dots,N_{l}-1\right\}$. Let $i$ be such that $i_{0}-2^{I_{l}+j+1}\leqslant i\leqslant i_{0}-2^{I_{l}+j}+1$ for some $j\in\left\{1,\dots,J_{l}\right\}$. We have

[TABLE]

Consequently, for any $k$ such that $2^{I_{l}+j-1}<k\leqslant 2^{I_{l}+j}$ and each $i$ such that $i_{0}-2^{I_{l}+j+1}\leqslant i\leqslant i_{0}-k-2^{I_{l}+j}+1$, we have

[TABLE]

It thus follows that

[TABLE]

and using disjointness of the sets $\mathsf{T}^{N_{l}-i_{0}}C_{l}$, $K_{l}\leqslant i_{0}\leqslant N_{l}-1$, we infer that

[TABLE]

where

[TABLE]

Since $\mathsf{T}$ is measure-preserving, the events $A_{i_{0}}\cap\mathsf{T}^{N_{l}-i_{0}}C_{l},\quad K_{l}\leqslant i_{0}\leqslant N_{l}-1$ have the same probability, which is equal to $\mathsf{P}(A_{K_{l}}\cap\mathsf{T}^{N_{l}-K_{l}}C_{l})$. The events $A_{K_{l}}$ and $\mathsf{T}^{N_{l}-K_{l}}C_{l}$ belong respectively to $\mathcal{B}$ and $\mathcal{C}$, hence they are independent. In view of (3.4.7), we obtain

[TABLE]

Now, in order to control the latter term, we shall use the following lemma:

Lemma 3.10**.**

Let $\left(H_{l}\right)_{l\geqslant 1}$ be an increasing sequence of integers. Assume that for each $l\geqslant 1$, the family of events $\left(A_{l,j}\right)_{1\leqslant j\leqslant H_{l}}$ is independent and that $\sum_{j=1}^{H_{l}}\mathsf{P}\left(A_{l,j}\right)\in[3/4,1]$. Then for each $l\geqslant 1$,

[TABLE]

Proof of Lemma 3.10.

By Bonferroni’s inequality, we have for any $l\geqslant 1$,

[TABLE]

Using independence of $\left(A_{l,j}\right)_{1\leqslant j\leqslant H_{l}}$, we derive that

[TABLE]

∎

We now use Lemma 3.10 with the choices $H_{l}=J_{l}$ and

[TABLE]

We indeed have, with the notations of (5.0.1) and by (3.4.2),

[TABLE]

hence by (3.4.1),

[TABLE]

We get, in view of (3.4.8) the lower bound

[TABLE]

This concludes the proof of Lemma 3.9.

Now, we prove that for any $l\geqslant 1$,

[TABLE]

We first prove that

[TABLE]

where $f^{\prime}_{l}:=\sum_{i=1}^{l}gf_{i}$. First note that for $1\leqslant k\leqslant K_{l}$ and $0\leqslant i\leqslant N_{l}-k$,

[TABLE]

hence

[TABLE]

Now, by definition of $f_{u}$, for each $\omega\in\Omega$, the following inequality holds: $\left|f_{u}\left(\omega\right)\right|\leqslant\left(\frac{N_{u}}{2^{I_{u}}}\right)^{1/q(p,\alpha)}$.

Consequently,

[TABLE]

and this term does not exceed $1$ by (3.4.3). This proves (3.4.10)

Now, defining $f^{\prime\prime}_{l}:=\sum_{u=l+1}^{+\infty}gf_{u}$, we have

[TABLE]

By constructing of $f_{u}$, we have \mathsf{P}\left(\Big{(}f_{u}\neq 0\Big{)}\right)\leqslant K_{u}/N_{u}, hence

[TABLE]

by (3.4.4).

Thus (3.4.9) follows from the combination of Lemma 3.9, (3.4.10) and (3.4.11). This ends the proof of Theorem 1.3.

4. Proofs: the case $\alpha\leqslant 1/p$

We start with the following lemma which reduces the proof of convergence to that of tightness.

Lemma 4.1**.**

Let $p\geqslant 1$ and $0\leqslant\alpha\leqslant\min\{1/2,1/p\}$. Assume that $Z$ is a random element in $B^{o}_{p,\alpha}$. Then for any stationary sequence $\left(f\circ\mathsf{T}^{j}\right)$ if

• $(i)$

$W_{f,n}\xrightarrow[n\to\infty]{\mathcal{D}}Z$* in $L_{p}[0,1]$, and*

• $(ii)$

$(W_{f,n})$* is tight in $B^{o}_{p,\alpha}$,*

then $W_{f,n}\xrightarrow[n\to\infty]{\mathcal{D}}Z$ in $B^{o}_{p,\alpha}$.

Proof.

From (ii) we have that each subsequence of $(W_{f,n})$ has further subsequence that converges in distribution. If $W_{f,n^{\prime}}\xrightarrow[n\to\infty]{\mathcal{D}}Y^{\prime}$ and $W_{f,n^{\prime\prime}}\xrightarrow[n\to\infty]{\mathcal{D}}Y^{\prime\prime}$ then we have that for Franklin basis $(f_{k})$ it holds that $\langle W_{f,n^{\prime}},f_{k}\rangle\xrightarrow[n\to\infty]{\mathcal{D}}\langle Y^{\prime},f_{k}\rangle$ and $\langle W_{f,n^{\prime\prime}},f_{k}\rangle\xrightarrow[n\to\infty]{\mathcal{D}}\langle Y^{\prime\prime},f_{k}\rangle$ for any $k$. But (i) gives that both $\langle Y^{\prime},f_{k}\rangle$ and $\langle Y^{\prime\prime},f_{k}\rangle$ have the same distribution as $\langle Z,f_{k}\rangle$. Since coefficients $\langle Z,f_{k}\rangle$ determines the distribution of $Z$ we can conclude that $Y^{\prime}$ and $Y^{\prime\prime}$ are equally distributed with $Z$. This ends the proof. ∎

4.1. Proof of Theorem 1.4

Due to continuity of the embedding $B^{o}_{2,\alpha}\hookrightarrow B^{o}_{p,\alpha}$ if $1\leqslant p\leqslant 2$ and $0\leqslant\alpha<1/2$. it is enough to prove the case where either $p=2$ and $0\leqslant\alpha<1/2$ or $p>2$ and $0\leqslant\alpha\leqslant 1/p$.

Recall $W_{f,n}=n^{-1/2}\zeta_{f,n}$. We shall prove for each $\varepsilon>0$

[TABLE]

where

[TABLE]

Since the function $W_{f,n}(t),0\leqslant t\leqslant 1$ is affine in each interval $\left((k-1)/n,k/n\right]$, it holds for $s,t\in[(k-1)/n,k/n],$

[TABLE]

This observation leads to

[TABLE]

where $c_{p}>0$ is a constant depending on $p$ only,

[TABLE]

As a consequence in order to establish (4.1.1) we have to prove

[TABLE]

Consider first (4.1.3) and start with $p=2$ and $0\leqslant\alpha<1/2$. By Chebyshev inequality

[TABLE]

and (4.1.3) follows in this case. Now let $p>2$ and $0\leqslant\alpha\leqslant 1/p$. For this case we shall use truncation. Set for $\tau>0$,

[TABLE]

Then $\mathsf{P}(\delta^{-\alpha}U_{f,n}(\delta)>\varepsilon)\leq n\mathsf{P}(|f|\geq\tau\sqrt{\max\{n,\delta^{-1}\}})+\mathsf{P}(\delta^{-\alpha}U_{f^{\prime},n}(\delta)>\varepsilon)$ and, since

[TABLE]

we reduce the proof of (4.1.3) to

[TABLE]

We have by Chebyshev inequality,

[TABLE]

Hence

[TABLE]

Since $\tau>0$ is arbitrary, the limit is indeed zero, and the proof of (4.1.5) is completed.

To prove (4.1.4) we start again with the case $p=2$ and $0\leqslant\alpha<1/2$. In this case Chebyshev inequality along with stationarity and Doob-Kolmogorov inequality yields

[TABLE]

and (4.1.4) follows. This ends the proof of (4.1.4) in the case $p=2$.

Assume that $p>2$ and $\alpha\leqslant 1/p$. Let us fix $\varepsilon>0$. Define for any $\delta\in(0,1)$ and any integer $n\geqslant 1/\delta$ the events

[TABLE]

[TABLE]

where $\tau$ is an arbitrary but fixed positive number. We have the bound

[TABLE]

Since the sequence $\left(\zeta_{f,n}\right)$ is tight in the space $C[0,1]$ (see [1]), we have

[TABLE]

Now, note that on $A_{n,j}\cap B_{n,j,\tau}^{c}$, we have for any $0\leqslant\ell\leqslant\lfloor n\delta\rfloor$,

[TABLE]

Therefore, we have

[TABLE]

and we infer that

[TABLE]

By Markov’s inequality, stationarity and Doob’s inequality, we have

[TABLE]

Since $p\alpha\leqslant 1$, we get

[TABLE]

and since $\tau$ is arbitrary, we get

[TABLE]

in view of (4.1.6). This concludes the proof of (4.1.4) and that of Theorem 1.4.

4.2. Proof of Theorem 1.5

It follows from Theorem 1.4 and the same arguments as used in the proof of Theorem 1.2.

5. Some applications

As already was mentioned in the introduction, a choice of functional spaces for polygonal line processes is usually inspired by possible applications in statistics via continuous mappings: if $W_{f,n}\xrightarrow[n\to\infty]{\mathcal{D}}W$ in the space $B^{o}_{p,\alpha}$, then $T(W_{f,n})\to T(W)$ for any continuous function $T:B^{o}_{p,\alpha}\to\mathbb{R}$. This general observation can be used, for example, to analyse so called $k$-scan processes

[TABLE]

Proposition 5.1**.**

Let $f$ be a function such that the sequence $\left(W_{n}(f)\right)_{n\geqslant 1}$ converges to a standard Brownian motion $W$ in $B_{p,\alpha}^{o}$, where $1/p<\alpha<1/2$. For each $a,b\in[0,1]$ such that $a<b$, we define

[TABLE]

Then the following convergence holds:

[TABLE]

Proof.

Let use define a functional $F\colon B_{p,\alpha}^{o}\to\mathbb{R}$ by

[TABLE]

Then $F$ is continuous with respect to the topology of $B_{p,\alpha}^{o}$ and $F(W)=Y_{a,b}$. We thus have $F\left(W_{n}\left(f\right)\right)\xrightarrow[n\to\infty]{\mathcal{D}}Y_{a,b}$. To conclude that (5.0.1) holds, it suffices to prove that

[TABLE]

First note that $Z_{n}$ is non-negative. Second, we have

[TABLE]

Let $k$ be an integer such that $\lfloor an\rfloor\leqslant k\leqslant\lfloor bn\rfloor+1$ and let $t$ be a real number such that $k/n\leqslant t<(k+1)/n$. Then

[TABLE]

which implies that

[TABLE]

By Theorem 2.4, the second term in the right hand side of (5.0.2) goes to zero in probability. Therefore, it suffices to prove that

[TABLE]

To see this, we start from the inequality

[TABLE]

Let $N$ be a fixed integer. The functional

[TABLE]

is continuous. Therefore, if $\varepsilon>0$ is a continuity point of the cumulative distribution function of $G_{N}(W)$ for each $N$, we have

[TABLE]

Now, take $q>p$. Using Hölder’s inequality with the exponents $q/p$ and $q/(q-p)$, we have

[TABLE]

Since $\lVert W\rVert_{q,\alpha}$ is almost surely finite, we get from (5.0.4) that (5.0.3) holds. ∎

Statistics based on $k$-scan processes can be used to detect epidemic change in the mean of a sample of size $n$ (see, e.g., [11], and reference therein). More precisely, given a sample $X_{1},X_{2},\dots,X_{n}$, consider the model

[TABLE]

where $(\varepsilon_{i})$ is a stationary sequence, $\mu\not=0$ and $k^{*},m^{*}$ are unknown parameters of the model. We want to test the null hypothesis $H_{0}:\ \ \mu=0$ against the alternative $\mu\not=0$. To this aim one can consider the statistics

[TABLE]

Under null its limit is defined by Proposition 5.1 provided $(\varepsilon_{i})$ satisfies the weak invariance principle in the Besov space $B^{o}_{p,\alpha}$. Under alternative then we see that

[TABLE]

as $n\to\infty$, where $h^{*}=m^{*}-k^{*}$ is the duration of the epidemic state.