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Rates for the CLT via new ideal metrics. (English) Zbl 0675.60018

Let \((B,\| \cdot \|)\) be a separable Banach space, \(X=X(B)\) the vector space of all random variables taking values in B. New ideal probability metrics of convolution type for the space X are introduced and it is shown that they provide refined rates of convergence of the sum \(S_ n=n^{-1/\alpha}(X_ 1+...+X_ n)\) of i.i.d. random variables in X(B) to a stable limit law \(Y_{\alpha}\) in X(B), where \(\alpha\in (0,2]\).
Reviewer: N.Leonenko

MSC:

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
60E07 Infinitely divisible distributions; stable distributions
60B10 Convergence of probability measures
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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