Approximation by infinitely divisible distributions in the multidimensional case

Abstract

Let

be the collection of parallelepipeds in R^κ with edges parallel with the coordinate axes and let

be the collection of closed sets in R^κ. Let π(G, H)=inf {ε∣G{A}⩽H{A^ε}+ε, H{A}⩽G{A^ε}+ε for any

; L(G, H)= inf {ε∣G{A}⩽H{A^ε}+ε, H{A}⩽G{A^ε}+ε for any

, where G, H are distributions in$R^\kappa ,A^\varepsilon = \left\{ {x \in R^\kappa \left| {\mathop {\inf }\limits_{y \in A} } \right.\left\| {x - y} \right\|< \varepsilon } \right\}$. In the paper one gives the proofs of results announced earlier by the author (Dokl. Akad. Nauk SSSR,253, No. 2, 277–279 (1980)). One considers the problem of the approximation of the distributions of sums of independent random vectors with the aid of infinitely divisible distributions. One obtains estimates for the distances π(·, ·), L(·, ·) and

. It is proved that

, where 0⩽p_i⩽1,$p = \mathop {\max }\limits_{1 \leqslant i \leqslant n} p_i$; E is the distribution concentrated at zero; V_i(i=1, ..., n) are arbitrary distributions; the products and the exponentials are understood in the sense of convolution.