Exponential deficiency of convolutions of densities

Document Type

Article

Publication Date

1-1-2012

Abstract

If a probability density p(x) (x ϵ ℝk) is bounded and R(t) := ∫ e⟨x, tu⟩ p(x)dx < ∞ for some linear functional u and all t ϵ (0, 1), then, for each t ϵ (0, 1) and all large enough n, the n-fold convolution of the t-tilted density p−t(x) := 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. © EDP Sciences, SMAI 2012.

Publication Title

ESAIM - Probability and Statistics

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