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

Author

Listed:
  • F. Götze

    (Universität Bielefeld)

  • A. Yu. Zaitsev

    (St. Petersburg Branch of Steklov Mathematical Institute)

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).

Suggested Citation

  • F. Götze & A. Yu. Zaitsev, 1998. "Estimates for the Rapid Decay of Concentration Functions of n-Fold Convolutions," Journal of Theoretical Probability, Springer, vol. 11(3), pages 715-731, July.
    • Handle: RePEc:spr:jotpro:v:11:y:1998:i:3:d:10.1023_a:1022654631571
    DOI: 10.1023/A:1022654631571
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