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).
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Götze, F., Zaitsev, A.Y. Estimates for the Rapid Decay of Concentration Functions of n-Fold Convolutions. Journal of Theoretical Probability 11, 715–731 (1998). https://doi.org/10.1023/A:1022654631571
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DOI: https://doi.org/10.1023/A:1022654631571