Exponential deficiency of convolutions of densities∗
Department of Mathematical Sciences, Michigan Technological University, Houghton, 49931 Michigan, USA
Received: 17 June 2009
Revised: 17 February 2010
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 := 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.
Mathematics Subject Classification: 60E05 / 60E10 / 60F10 / 62E20 / 60E15
Key words: Probability density / saddle-point approximation / sums of independent random variables/vectors / convolution / exponential integrability / boundedness / exponential tilting / exponential families / absolute integrability / characteristic functions
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