On limit theorems for fields of martingale differences

2018/03/24 by Dalibor Volny, Volny, Dalibor
Mathematics · #28D15 #60F05 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:28D15 #msc:60F05

paper · pdf · doi:10.48550/arxiv.1803.09100

arxiv created 2018/03/24 · arxiv updated 2018/03/28

Abstract

We prove a central limit theorem for stationary multiple (random) fields of martingale differences f∘ T_\underlinei, \underlinei∈ \Bbb Z^d, where T_\underlinei 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≥ 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 ∑_i X_n,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.

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