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Exponential deficiency of convolutions of densities

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 \hbox{$\tilde p_t$}˜pt := 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.

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 \hbox{$\tilde p_t$}˜pt := 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.

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H.E. Daniels , Tail probability approximations. Int. Stat. Rev. 55 (1987) 3748.

P. Embrechts and C.M. Goldie , On convolution tails. Stoch. Proc. Appl. 13 (1982) 263278.

B.-Y. Jing , Q.-M. Shao and W. Zhou , Saddlepoint approximation for Student’s t-statistic with no moment conditions. Ann. Stat. 32 (2004) 26792711.

C. Klüppelberg , Subexponential distributions and characterizations of related classes. Probab. Theory Relat. Fields 82 (1989) 259269.

R. Lugannani and S. Rice , Saddle point approximation for the distribution of the sum of independent random variables. Adv. Appl. Probab. 12 (1980) 475490.

N. Reid , Saddlepoint methods and statistical inference. Stat. Sci. 3 (1988) 213238. With comments and a rejoinder by the author.

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ESAIM: Probability and Statistics
  • ISSN: 1292-8100
  • EISSN: 1262-3318
  • URL: /core/journals/esaim-probability-and-statistics
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