Exponential deficiency of convolutions of densities

Abstract

If a probability density p(x) (x 2 R k ) is bounded and R(t) := R e h x,tui p(x) dx < 1 for some linear functional u and all t 2 (0, 1), then, for each t 2 (0, 1) and all large enough n, the n-fold convolution of the t- tilted density ˜ pt(x) := e h x,tui 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 are useful for saddle-point approximations. AMS 2000 subject classifications: Primary 60E05, 60E10; secondary 60F10, 62E20, 60E15. Keywords and phrases: probability density, saddle-point approximation, sums of independent random variables/vectors, convolution, exponential integrability, boundedness, tilting, exponential families.