On limit theorems for fields of martingale differences

Abstract

We prove a central limit theorem for stationary multiple (random) fields of martingale differences f∘Ti̲, i̲∈Zd, where Ti̲ is a Zd 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 Volný (2015) 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 Zd action is ergodic and d≥2, the limit law need not be normal.For proving the result mentioned above, a generalisation of McLeish’s CLT for arrays (Xn,i) of martingale differences is used. More precisely, sufficient conditions for a CLT are found in the case when the sums ∑iXn,i2 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.