Exponential deficiency of convolutions of densities∗

Iosif Pinelis

ESAIM: Probability and Statistics (2012)

  • Volume: 16, page 86-96
  • ISSN: 1292-8100

Abstract

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If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫e〈x, tu〉p(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 p ˜ t := e〈x, tu〉p(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.

How to cite

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Pinelis, Iosif. "Exponential deficiency of convolutions of densities∗." ESAIM: Probability and Statistics 16 (2012): 86-96. <http://eudml.org/doc/222476>.

@article{Pinelis2012,
abstract = {If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫e〈x, tu〉p(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\{$\tilde p_t$\} := e〈x, tu〉p(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. },
author = {Pinelis, Iosif},
journal = {ESAIM: Probability and Statistics},
keywords = {Probability density; saddle-point approximation; sums of independent random variables/vectors; convolution; exponential integrability; boundedness; exponential tilting; exponential families; absolute integrability; characteristic functions; probability density},
language = {eng},
month = {7},
pages = {86-96},
publisher = {EDP Sciences},
title = {Exponential deficiency of convolutions of densities∗},
url = {http://eudml.org/doc/222476},
volume = {16},
year = {2012},
}

TY - JOUR
AU - Pinelis, Iosif
TI - Exponential deficiency of convolutions of densities∗
JO - ESAIM: Probability and Statistics
DA - 2012/7//
PB - EDP Sciences
VL - 16
SP - 86
EP - 96
AB - If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫e〈x, tu〉p(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{$\tilde p_t$} := e〈x, tu〉p(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.
LA - eng
KW - Probability density; saddle-point approximation; sums of independent random variables/vectors; convolution; exponential integrability; boundedness; exponential tilting; exponential families; absolute integrability; characteristic functions; probability density
UR - http://eudml.org/doc/222476
ER -

References

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  6. C. Klüppelberg, Subexponential distributions and characterizations of related classes. Probab. Theory Relat. Fields82 (1989) 259–269.  
  7. R. Lugannani and S. Rice, Saddle point approximation for the distribution of the sum of independent random variables. Adv. Appl. Probab.12 (1980) 475–490.  
  8. I.F. Pinelis, Asymptotic equivalence of the probabilities of large deviations for sums and maximum of independent random variables, in Limit theorems of probability theory. “Nauka” Sibirsk. Otdel., Novosibirsk. Trudy Inst. Mat.5 (1985) 144–173, 176.  
  9. N. Reid, Saddlepoint methods and statistical inference. Stat. Sci.3 (1988) 213–238. With comments and a rejoinder by the author. 
  10. Q.-M. Shao, Recent progress on self-normalized limit theorems, in Probability, finance and insurance. World Sci. Publ., River Edge, NJ (2004) 50–68.  

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