Estimates for the Rapid Decay of Concentration Functions of n-Fold Convolutions

F. Götze () and A. Yu. Zaitsev ()
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F. Götze: Universität Bielefeld
A. Yu. Zaitsev: St. Petersburg Branch of Steklov Mathematical Institute

Journal of Theoretical Probability, 1998, vol. 11, issue 3, 715-731

Abstract: Abstract We estimate the concentration functions of n-fold convolutions of one-dimensional probability measures. The Kolmogorov–Rogozin inequality implies that for nondegenerate distributions these functions decrease at least as O(n −1/2). On the other hand, Esseen(3) has shown that this rate is o(n −1/2) iff the distribution has an infinite second moment. This statement was sharpened by Morozova.(9) Theorem 1 of this paper provides an improvement of Morozova's result. Moreover, we present more general estimates which imply the rates o(n −1/2).

Keywords: Concentration functions; sums of i.i.d. random variables; rates of decay; n-fold convolutions (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1023/A:1022654631571

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