On limit theorems for fields of martingale differences

TL;DR

This paper establishes a general central limit theorem for stationary fields of martingale differences indexed by multi-dimensional integer lattices, showing convergence to a mixture of normal laws and extending previous results to non-ergodic cases.

Contribution

It generalizes CLTs for multi-dimensional martingale difference fields, including non-ergodic cases, and introduces new conditions for convergence when sums of squares converge in distribution.

Findings

Convergence to a mixture of normal laws in general cases.

CLT and invariance principles hold in non-ergodic settings.

Extension of McLeish's CLT for arrays of martingale differences.

Abstract

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.