Exponential deficiency of convolutions of densities

Pinelis, Iosif 

ESAIM: Probability and Statistics, Tome 16 (2012), pp. 86-96

Résumé

If a probability density p(x) (x ∈ ℝ^k) 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.

Détail

DOI : 10.1051/ps/2010010

Classification : 60E05, 60E10, 60F10, 62E20, 60E15
Keywords: probability density, saddle-point approximation, sums of independent random variables/vectors, convolution, exponential integrability, boundedness, exponential tilting, exponential families, absolute integrability, characteristic functions