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 ˜pt := 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/274356>.

@article{Pinelis2012,
abstract = {If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫e〈x, tu〉p(x)dx &lt; ∞ 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 $\tilde\{p\}_t$˜pt := 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},
language = {eng},
pages = {86-96},
publisher = {EDP-Sciences},
title = {Exponential deficiency of convolutions of densities},
url = {http://eudml.org/doc/274356},
volume = {16},
year = {2012},
}

TY - JOUR
AU - Pinelis, Iosif
TI - Exponential deficiency of convolutions of densities
JO - ESAIM: Probability and Statistics
PY - 2012
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 &lt; ∞ 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 $\tilde{p}_t$˜pt := 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
UR - http://eudml.org/doc/274356
ER -

References

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  7. [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. Zbl0425.60042MR569438
  8. [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. Zbl0607.60023MR821760
  9. [9] N. Reid, Saddlepoint methods and statistical inference. Stat. Sci. 3 (1988) 213–238. With comments and a rejoinder by the author. Zbl0955.62541MR968390
  10. [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. MR2189198

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