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

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.Comment: Stochastics and Dynamics 201

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.Comment: 17 page

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

In 1983, N. Herrndorf proved that for a $\phi$-mixing sequence satisfying the central limit theorem and $\liminf_{n\to\infty}\frac{\sigma^2_n}n>0$, the weak invariance principle takes place. The question whether for strictly stationary sequences with finite second moments and a weaker type ($\alpha$, $\beta$, $\rho$) of mixing the central limit theorem implies the weak invariance principle remained open. We construct a strictly stationary $\beta$-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.Comment: 12 page

On the central and local limit theorem for martingale difference sequences

Let (\Omega, \A, \mu) be a Lebesgue space and $T$ an ergodic measure preserving automorphism on $\Omega$ with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on $\Omega$ 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.Comment: Accepte pour publication dans Stochastics and Dynamic

Stable CLT for deterministic systems

We show that for every ergodic and aperiodic probability preserving transformation and $\alpha\in (0,2)$ there exists a function whose associated time series is in the standard domain of attraction of a non-degenerate symmetric $\alpha$-stable distribution.Comment: 17 pages, 0 figure

LOCAL LIMIT THEOREM IN DETERMINISTIC SYSTEMS

We show that for every ergodic and aperiodic probability preserving system, there exists a Z valued, square integrable function f such that the partial sums process of the time series f • T i ∞ i=0 satisfies the lattice local limit theorem