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Exponential deficiency of convolutions of densities∗

Published online by Cambridge University Press:  02 July 2012

Iosif Pinelis
Iosif Pinelis*
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, Houghton, 49931 Michigan, USA. ipinelis@mtu.edu
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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{$\tilde p_t$}˜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.

Keywords

Probability densitysaddle-point approximationsums of independent random variables/vectorsconvolutionexponential integrabilityboundednessexponential tiltingexponential familiesabsolute integrabilitycharacteristic functions
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 86 - 96
Copyright
© EDP Sciences, SMAI, 2012

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References

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