Approximation of distributions of sums of Banach-valued random elements with infinitely divisible laws. I.
The authors investigate the approximation of distributions of sums of i.i.d. random vectors in a Banach space with infinite divisible laws. They consider general questions of weak approximation and present some examples. Under the assumption that the Banach space B is of finite q- cotype, \(1\leq q<\infty\), they find estimates of the closeness of distributions of sums of i.i.d. symmetric random vectors to the class of moving laws in the Prokhorov metric. Let \(\gamma_ n(\alpha)=\inf \{t:P(\| S_ n\| \leq t)\geq \alpha \}\), \(0<\alpha <1\), \(n\geq 1\), then the main result is as follows: For any \(\alpha\) such that \(1-3^{-q}/8<\alpha <1\), there exists a constant \(C=C(B,q,\alpha)\) such that for any symmetric random variables \(X_ 1,X_ 2,..\). with values in B one has \[ \pi (S_ n/\gamma_ n(\alpha);\quad S_{\nu_ n}/\gamma_ n(\alpha))\leq C((\ln (n+1))/n)^{1/[2(q+1)]}, \] where \(\pi\) is the Prokhorov metric and \(\nu_ n\) is a Poisson random variable with parameter n, independent of \(X_ 1,X_ 2,... \).
scientific article; zbMATH DE number 4020078
scientific article