# Exponential deficiency of convolutions of densities

ESAIM: Probability and Statistics (2012)

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

## Access Full Article

top## Abstract

top## How to cite

topPinelis, 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 < ∞ 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 < ∞ 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

top- [1] O. Barndorff-Nielsen and D.R. Cox, Edgeworth and saddle-point approximations with statistical applications. J. R. Stat. Soc., Ser. B 41 (1979) 279–312. With discussion. Zbl0424.62010MR557595
- [2] R.N. Bhattacharya and R.R. Rao, Normal approximation and asymptotic expansions. Robert E. Krieger Publishing Co. Inc., Melbourne, FL (1986). Reprint of the 1976 original. Zbl0331.41023MR855460
- [3] H.E. Daniels, Tail probability approximations. Int. Stat. Rev.55 (1987) 37–48. Zbl0614.62016MR962940
- [4] P. Embrechts and C.M. Goldie, On convolution tails. Stoch. Proc. Appl.13 (1982) 263–278. Zbl0487.60016MR671036
- [5] B.-Y. Jing, Q.-M. Shao and W. Zhou, Saddlepoint approximation for Student’s t-statistic with no moment conditions. Ann. Stat.32 (2004) 2679–2711. Zbl1068.62016MR2153999
- [6] C. Klüppelberg, Subexponential distributions and characterizations of related classes. Probab. Theory Relat. Fields82 (1989) 259–269. Zbl0687.60017MR998934
- [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] 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] N. Reid, Saddlepoint methods and statistical inference. Stat. Sci. 3 (1988) 213–238. With comments and a rejoinder by the author. Zbl0955.62541MR968390
- [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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.