arXiv subjects

Dalibor Volny

Publications and source records attributed to Dalibor Volny.

17 recordsLinked to original sources

arXiv2024

What can be the limit in the CLT for a field of martingale differences?

The now classical convergence in distribution theorem for well normalized sums ofstationary martingale increments has been extended to multi-indexed martingaleincrements (see Volný (2019) and references in there). In the presentarticle we make progress in the identification of the limit law.In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is stillthe case for multi-indexed martingale increments when one of the processes defined by one coordinate of the{\it multidimensional time} is ergodic. In the general case, the limit may be non normal.The dynamical properties of the $\mathbb{Z}^d$-measure preserving action associatedto the stationary random field allows us to give a necessary and sufficient conditionfor the existence of a non-normal limit law, in terms of entropy of some random processes.The identification of a {\it natural} factor on which the $\mathbb{Z}^d$-action is {\it of product type

math.DS↗

arXiv2023

Martingale approximation of non-stationary stochastic processes

We generalise the martingale-coboundary representation of discrete time stochastic processes to the non-stationary case and to random variables in Orlicz spaces. Related limit theorems (CLT, invariance principle, log log law, probabilities of large deviations) are studied.

math.PR↗

arXiv2023

Some remarks on the ergodic theorem for $U$-statistics

In this note, we investigate the convergence of a $U$-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and $L^1$ convergence and present some counter-examples showing that the $U$-statistic itself might fail to converge: centering is needed as well as boundedness of $\sup_{j\geq 2}\mathbb{E}[|h(X_1,X_j)|]$.

math.PR↗

arXiv2023

Stable CLT for deterministic systems

We show that for every ergodic and aperiodic probability preserving transformation and $α\in (0,2)$ there exists a function whose associated time series is in the standard domain of attraction of a non-degenerate symmetric $α$-stable distribution.

math.DS↗

arXiv2019

Local limit theorem in deterministic systems

We show that for every ergodic and aperiodic probability preserving system, there exists a $\mathbb{Z}$ valued, square integrable function $f$ such that the partial sums process of the time series $\left\{f\circ T^i\right\}_{i=0}^\infty$ satisfies the lattice local limit theorem.

math.DS↗

arXiv2018

On limit theorems for fields of martingale differences

We prove a central limit theorem for stationary multiple (random) fields of martingale differences $f\circ T_{\underline{i}}$, $\underline{i}\in \Bbb Z^d$, where $T_{\underline{i}}$ is a $\Bbb Z^d$ action. In most cases the multiple (random) fields of martingale differences is given by a completely commuting filtration. A central limit theorem proving convergence to a normal law has been known for Bernoulli random fields and in [V15] this result was extended to random fields where one of generating transformations is ergodic. In the present paper it is proved that a convergence takes place always and the limit law is a mixture of normal laws. If the $\Bbb Z^d$ action is ergodic and $d\geq 2$, the limit law need not be normal. For proving the result mentioned above, a generalisation of McLeish's CLT for arrays $(X_{n,i})$ of martingale differences is used. More precisely, sufficient conditions for a CLT are found in the case when the sums $\sum_i X_{n,i}^2$ converge only in distribution. The CLT is followed by a weak invariance principle. It is shown that central limit theorems and invariance principles using martingale approximation remain valid in the non-ergodic case.

math.PR↗

arXiv2017

Martingale-coboundary decomposition for stationary random fields

We prove a martingale-coboundary representation for random fields with a completely commuting filtration. For random variables in L2 we present a necessary and sufficient condition which is a generalization of Heyde's condition for one dimensional processes from 1975. For Lp spaces with 2 \leq p < \infty we give a necessary and sufficient condition which extends Volny's result from 1993 to random fields and improves condition of El Machkouri and Giraudo from 2016 (arXiv:1410.3062). In application, new weak invariance principle and estimates of large deviations are found.

math.PR↗

arXiv2015

Quenched central limit theorems for a stationary linear process

We find a sufficient condition under which a central limit theorem for a stationary linear process is quenched. We find a stationary linear process szatisfying the Maxwell-Woodroofe condition for which the variances of partial sums are o(n), there is a CLT with a convergence towards N(0,1) when dividing by standard deviation of the partial sums, and the CLT is not quenched. The weak invariance principle does not hold.

math.PR↗

arXiv2015

A central limit theorem for fields of martingale differences

We prove a central limit theorem for a random field generated by d commuting probability preserving transformations; the martingale is given by a commuting filtration (cf. D. Khosnevisan, Multiparameter Processes, Springer 2002). The result has been known for Bernoulli random fields. Here, only ergodicity of one of generating transformations is supposed.

math.PR↗

arXiv2014

A strictly stationary $β$-mixing process satisfying the central limit theorem but not the weak invariance principle

In 1983, N. Herrndorf proved that for a $ϕ$-mixing sequence satisfying the central limit theorem and $\liminf_{n\to\infty}\frac{σ^2_n}n>0$, the weak invariance principle takes place. The question whether for strictly stationary sequences with finite second moments and a weaker type ($α$, $β$, $ρ$) of mixing the central limit theorem implies the weak invariance principle remained open. We construct a strictly stationary $β$-mixing sequence with finite moments of any order and linear variance for which the central limit theorem takes place but not the weak invariance principle.

math.PR↗

arXiv2012

A central limit theorem for stationary random fields

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form $X_k = g(\varepsilon_{k-s}, s \in \Z^d)$, $k\in\Z^d$, where $(\varepsilon_i)_{i\in\Z^d}$ are i.i.d random variables and $g$ is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying domain on which the process is observed. A limit theorem for the sample auto-covariance function is also established.

math.PR↗

arXiv2012

A quenched invariance principle for stationary processes

In this note, we prove a conditionally centered version of the quenched weak invariance principle under the Hannan condition, for stationary processes. In the course, we obtain a (new) construction of the fact that any stationary process may be seen as a functional of a Markov chain.

math.PR↗

arXiv2010

A central limit theorem for reversible processes with non-linear growth of variance

Kipnis and Varadhan showed that for an additive functional, $S_n$ say, of a reversible Markov chain the condition $E(S_n^{2})/n \to κ\in (0,\infty)$ implies the convergence of the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$, given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, $E(S_n^{2}) = n\ell(n)$, where $\ell$ is a slowly varying function. It is shown by example that the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$ need not converge to the standard normal distribution in this case; and sufficient conditions for convergence to a (possibly non-standard) normal distribution are developed.

math.PR↗

arXiv2009

An Indicator Function Limit Theorem in Dynamical Systems

We show by a constructive proof that in all aperiodic dynamical system, for all sequences $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a set $A\in\A$ having the property that the sequence of the distributions of $(\frac{1}{a_{n}}S_{n}(\ind_A-μ(A)))_{n\in\N}$ is dense in the space of all probability measures on $\R$.

math.DS↗

arXiv2009

On Sums of Indicator Functions in Dynamical Systems

In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every aperiodic dynamical system, for every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a measurable set $A$ such that the sequence of the distributions of the partial sums $\frac{1}{a_n}\sum_{i=0}^{n-1}(\ind_A-μ(A))\circ T^i$ is dense in the set of the probability measures on $\R$. Further, in the ergodic case, we prove that there exists a dense $G_δ$ of such sets.

math.DS↗

arXiv2004

A salad of cocycles

We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed. Smooth versions of some of the constructions are also given.

math.DS↗

arXiv2004

On the central and local limit theorem for martingale difference sequences

Let $(Ω, \A, μ)$ be a Lebesgue space and $T$ an ergodic measure preserving automorphism on $Ω$ with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on $Ω$ with a common non-degenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.

math.PR↗