ESAIM: Probability and Statistics

Research Article

Exponential deficiency of convolutions of densities 

Iosif Pinelis

Department of Mathematical Sciences, Michigan Technological University, Houghton, 49931 Michigan, USA. ipinelis@mtu.edu

Abstract

If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫ex, tup(x)dx < ∞ for some linear functional u and all t ∈ (0,1), then, for each t ∈ (0,1) and all large enough n, the n-fold convolution of the t-tilted density := ex, tup(x)/R(t) is bounded. This is a corollary of a general, “non-i.i.d.” result, which is also shown to enjoy a certain optimality property. Such results and their corollaries stated in terms of the absolute integrability of the corresponding characteristic functions are useful for saddle-point approximations.

(Received June 17 2009)

(Revised February 17 2010)

(Online publication July 02 2012)

Key Words:

  • Probability density;
  • saddle-point approximation;
  • sums of independent random variables/vectors;
  • convolution;
  • exponential integrability;
  • boundedness;
  • exponential tilting;
  • exponential families;
  • absolute integrability;
  • characteristic functions

Mathematics Subject Classification:

  • 60E05;
  • 60E10;
  • 60F10;
  • 62E20;
  • 60E15

Footnotes

  Supported by NSF grant DMS-0805946.

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