Exponential deficiency of convolutions of densities^∗

Department of Mathematical Sciences, Michigan Technological University, Houghton, 49931 Michigan, USA
ipinelis@mtu.edu

Received: 17 June 2009
Revised: 17 February 2010

Abstract

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 := 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.