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Dalibor Volný

Publications and source records attributed to Dalibor Volný.

12 recordsLinked to original sources
arXiv2023

Stable Functional CLT for deterministic systems

We show that alpha stable Lévy motions can be simulated by any ergodic and aperiodic probability preserving transformation. Namely we show: - for $0<α<1$ and every $α$ stable Lévy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$. - for $1\leq α<2$ and every symmetric alpha stable Lévy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$, - for $1< α<2$ and every $-1\leqβ\leq 1$ there exists a function f whose associated time series is in the classical domain of attraction of an $S_α(\ln(2), β,0)$ random variable.

math.DS↗
arXiv2019

Quenched invariance principles for orthomartingale-like sequences

In this paper we study the central limit theorem and its functional form for random fields which are not started from their equilibrium, but rather under the measure conditioned by the past sigma field. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not its quenched form. This example also clarifies the optimality of the moment conditions used for the validity of our results. Finally, by using the so called orthomartingale-coboundary decomposition, we apply our results to linear and nonlinear random fields.

math.PR↗
arXiv2018

Stable limits for Markov chains via the Principle of Conditioning

We study limit theorems for partial sums of instantaneous functions of a homogeneous Markov chain on a general state space. The summands are heavy-tailed and the limits are stable distributions. The conditions imposed on the transition operator $P$ of the Markov chain ensure that the limit is the same as if the summands were independent. Such a~scheme admits a physical interpretation, as given in Jara et al. (Ann. Appl. Probab., 19 (2009), 2270--2300). We considerably extend the results of Jara et al., (ibid.) and Cattiaux and Manou-Abi (ESAIM Probab. Stat., 18 (2014), 468--486). We show that the theory holds under the assumption of operator uniform integrability in $L^2$ of $P$ (a notion introduced by Wu (J. Funct. Anal., 172 (2000), 301--376)) plus the $L^2$-spectral gap property. If we strengthen the uniform integrability in $L^2$ to the hyperboundedness, then the $L^2$-spectral gap property can be relaxed to the strong mixing at geometric rate (in practice: to geometric ergodicity). We provide an example of a Markov chain on a countable space that is uniformly integrable in $L^2$ (and admits an $L^2$-spectral gap), while it is not hyperbounded. Moreover, we show by example that hyperboundedness is still a weaker property than $ϕ$-mixing, what enlarges the range of models of interest. What makes our assumptions working is a new, efficient version of the Principle of Conditioning that operates with conditional characteristic functions rather than predictable characteristics.

math.PR↗
arXiv2015

Limit theorems for weighted Bernoulli random fields under Hannan's condition

Consider a Bernoulli random field satisfying the Hannan's condition. Recently, invariance principles for partial sums of random fields over rectangular index sets are established. In this note we complement previous results by investigating limit theorems for weighted Bernoulli random fields, including central limit theorems for partial sums over arbitrary index sets and invariance principles for Gaussian random fields. Most results improve earlier ones on Bernoulli random fields under Wu's condition, which is stronger than Hannan's condition.

math.PR↗
arXiv2014

An invariance principle for stationary random fields under Hannan's condition

We establish an invariance principle for a general class of stationary random fields indexed by $\mathbb Z^d$, under Hannan's condition generalized to $\mathbb Z^d$. To do so we first establish a uniform integrability result for stationary orthomartingales, and second we establish a coboundary decomposition for certain stationary random fields. At last, we obtain an invariance principle by developing an orthomartingale approximation. Our invariance principle improves known results in the literature, and particularly we require only finite second moment.

math.PR↗
arXiv2011

On martingale approximation of adapted processes

We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to wich it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.

math.PR↗
arXiv2010

Quenched Central Limit Theorems for Sums of Stationary Processes

It is shown that the existence of an L^1 co boundary does not imply the quenched version of the central limit theorem. In another result it is shown that Hannan's condition does imply quenched convergence for an appropriately centered version of the sum.

math.PR↗
arXiv2009

Martingale approximation and optimality of some conditions for the central limit theorem

Let $(X_i)$ be a stationary and ergodic Markov chain with kernel $Q$, $f$ an $L^2$ function on its state space. If $Q$ is a normal operator and $f = (I-Q)^{1/2}g$ (which is equivalent to the convergence of $\sum_{n=1}^\infty \frac{\sum_{k=0}^{n-1}Q^kf}{n^{3/2}}$ in $L^2$), we have the central limit theorem (cf\. \cite{D-L 1}, \cite{G-L 2}). Without assuming normality of $Q$, the CLT is implied by the convergence of $\sum_{n=1}^\infty \frac{\|\sum_{k=0}^{n-1}Q^kf\|_2}{n^{3/2}}$, in particular by $\|\sum_{k=0}^{n-1}Q^kf\|_2 = o(\sqrt n/\log^q n)$, $q>1$ by \cite{M-Wu} and \cite{Wu-Wo} respectively. We shall show that if $Q$ is not normal and $f\in (I-Q)^{1/2} L^2$, or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to $\sum_{n=1}^\infty c_n\frac{\|\sum_{k=0}^{n-1}Q^kf\|_2}{n^{3/2}}<\infty$ for some sequence $c_n\searrow 0$, or by $\|\sum_{k=0}^{n-1}Q^kf\|_2 = O(\sqrt n/\log n)$, the CLT need not hold.

math.PR↗
arXiv2008

New Techniques for Empirical Process of Dependent Data

We present a new technique for proving empirical process invariance principle for stationary processes $(X_n)_{n\geq 0}$. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions $(f(X_n))_{n\geq 0}$, not containing the indicator functions. Our approach can be applied to Markov chains and dynamical systems, using spectral properties of the transfer operator. Our proof consists of a novel application of chaining techniques.

math.PR↗
arXiv2008

Comparison between criteria leading to the weak invariance principle

The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields 110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in every ergodic dynamical system with positive entropy, if we consider two of these criteria, we can find a function in $\mathbb{L}^2$ satisfying the first but not the second.

math.PR↗
arXiv2008

Exactness of martingale approximation and the central limit theorem

The article is showing sharpness of central limit theorems of Kipnis and Varadhan, Derriennic and Lin, Maxwell and Woodroofe. In the case of the CLT of Derriennic and Lin (for Markov chains with a normal operator) it is shown that the assumption of normality cannot be relaxed. In the case of the CLT of Maxwell and Woodroofe, the example of Peligrad and Utev is improved in the sense of getting a convergence to different laws.

math.PR↗