Dalibor Volný

Large deviations for martingales

Let (Xi) be a martingale difference sequence and Sn=[summation operator]i=1n Xi. We prove that if supi E(eXi) 0 such that [mu](Sn>n)[less-than-or-equals, slant]e-cn1/3; this bound is optimal for the class of martingale difference sequences which are also strictly stationary and ergodic. If the sequence (Xi) is bounded in Lp, 2[less-than-or-equals, slant]p n)[less-than-or-equals, slant]cn-p/2 which is again optimal for strictly stationary and ergodic sequences of martingale differences. These estimations can be extended to martingale difference fields. The results are also compared with those for iid sequences; we give a simple proof that the estimate of Nagaev, Baum and Katz, [mu](Sn>n)=o(n1-p) for Xi[set membership, variant]Lp, 1[less-than-or-equals, slant]p 0.

Invariance principles and Gaussian approximation for strictly stationary processes

We show that in any aperiodic and ergodic dynamical system there exists a square integrable process (f ◦ T i) the partial sums of which can be closely approximated by the partial sums of Gaussian i.i.d. random variables. For (f ◦ T i) both weak and strong invariance principles hold.

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. Poincar{e} 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

Sums of continuous and differentiable functions in dynamical systems

mathbb{L}^2

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

satisfying the first but not the second.

Central Limit Theorems For Superlinear Processes

LetT be a homeomorphism of a metrizable compactX, the sequenceck/k tends to 0 andck tends to infinity. We’ll study the limit behaviour of the distributions of the sums (1/ck) ∑i=0k-1F oTi whereF is from a space of continuous functions—the central limit problem and the speed of convergence in the ergodic theorem.The main attention is given to the case whereX is the unit circle andT is an irrational rotation; in this case we consider the spaces of absolutely continuous, Lipschitz, andk-times differentiable functionsF.

Random ergodic theorems and real cocycles

In 1983, N. Herrndorf proved that for a ϕ-mixing sequence satisfying the central limit theorem and lim infn→∞σn2/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.

Contre-exemple dans le théorème central limite fonctionnel pour les champs aléatoires réels

The Central Limit Theorem is studied for stationary sequences that are sums of countable collections of linear processes. Two sets of sufficient conditions are obtained. One restricts only the coefficients and is shown to be best possible among such conditions. The other involves an interplay between the coefficients and the distribution functions of the innovations and is shown to be necessary for the Conditional Central Limit Theorem in the case of a causal process with independent innovations.

On Sums of Indicator Functions in Dynamical Systems

AbstractWe study mean convergence of ergodic averagesn