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Asymptotic expansions in the central limit theorem for a special class of m-dependent random fields. II: Lattice case. (English) Zbl 0705.60025
The paper is concerned with the following type of finitely dependent d- dimensionally indexed random fields: $X_ z=f_ z(\xi_ y;\quad y\in \times^{d}_{i=1}\{z_ i,...,z_ i+m_ i(n)\})$ for $$z=(z_ 1,...,z_ d)\in V_ n=\times^{d}_{i=1}\{1,2,...,n_ i(n)\}$$, where $$(\xi_ y)$$ is a field of independent rv’s and $$(f_ z)$$ is a family of real-valued Borel measurable functions on $$R^ M$$ with $$M=(m_ 1(n)+1)...(m_ d(n)+1)$$. In the first part of this paper [ibid. 134, 83- 106 (1987; Zbl 0636.60021)] asymptotic expansions for the distribution function as well as for the density of the standardized sum $(S_ n-E S_ n)/(Var S_ n)^{1/2},\quad S_ n=X_ 1+...+X_ n,$ were obtained. In this second part, the corresponding question is treated in the lattice case, i.e. each of the $$X_ z's$$ takes only values on the integer lattice $$\{...,-1,0,+1,...\}$$. The proof of the results consists of two main steps:
Firstly, as in Part I, a Taylor expansion of the cf of $$S_ n$$ is derived. Secondly, conditions are imposed on the “conditional lattices” of the individual rv’s $$X_ z$$ given $$\{\xi_ y$$; $$0<\| z-y\| \leq p\}$$ for some $$p\geq 1$$, in order to obtain rates of convergence in the CLT comparable with those in the classical i.i.d. case. Some applications to several stochastic models are discussed.
Reviewer: L.Heinrich

##### MSC:
 60F05 Central limit and other weak theorems 60G60 Random fields
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