TY - JOUR

T1 - Approximation by infinitely divisible distributions in the multidimensional case

AU - Zaitsev, A. Yu

PY - 1984/12/1

Y1 - 1984/12/1

N2 - Let[Figure not available: see fulltext.] be the collection of parallelepipeds in Rκ with edges parallel with the coordinate axes and let[Figure not available: see fulltext.] be the collection of closed sets in Rκ. Let π(G, H)=inf {ε{divides}G{A}≤H{Aε}+ε, H{A}≤G{Aε}+ε for any[Figure not available: see fulltext.]; L(G, H)= inf {ε{divides}G{A}≤H{Aε}+ε, H{A}≤G{Aε}+ε for any[Figure not available: see fulltext.], where G, H are distributions in {Mathematical expression}. 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[Figure not available: see fulltext.]. It is proved that[Figure not available: see fulltext.], where 0≤pi≤1, {Mathematical expression}; E is the distribution concentrated at zero; Vi(i=1, ..., n) are arbitrary distributions; the products and the exponentials are understood in the sense of convolution.

AB - Let[Figure not available: see fulltext.] be the collection of parallelepipeds in Rκ with edges parallel with the coordinate axes and let[Figure not available: see fulltext.] be the collection of closed sets in Rκ. Let π(G, H)=inf {ε{divides}G{A}≤H{Aε}+ε, H{A}≤G{Aε}+ε for any[Figure not available: see fulltext.]; L(G, H)= inf {ε{divides}G{A}≤H{Aε}+ε, H{A}≤G{Aε}+ε for any[Figure not available: see fulltext.], where G, H are distributions in {Mathematical expression}. 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[Figure not available: see fulltext.]. It is proved that[Figure not available: see fulltext.], where 0≤pi≤1, {Mathematical expression}; E is the distribution concentrated at zero; Vi(i=1, ..., n) are arbitrary distributions; the products and the exponentials are understood in the sense of convolution.

UR - http://www.scopus.com/inward/record.url?scp=34250135011&partnerID=8YFLogxK

U2 - 10.1007/BF01850670

DO - 10.1007/BF01850670

M3 - Article

AN - SCOPUS:34250135011

VL - 27

SP - 3227

EP - 3237

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -