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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

60F05 Central limit and other weak theorems
60G60 Random fields
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