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.

Keywords: probability density, saddle-point approximation, sums of independent random variables/vectors, convolution, exponential integrability, boundedness, exponential tilting, exponential families, absolute integrability, characteristic functions

@article{PS_2012__16__86_0, author = {Pinelis, Iosif}, title = {Exponential deficiency of convolutions of densities}, journal = {ESAIM: Probability and Statistics}, pages = {86--96}, publisher = {EDP-Sciences}, volume = {16}, year = {2012}, doi = {10.1051/ps/2010010}, mrnumber = {2946121}, zbl = {1266.60021}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps/2010010/} }

Pinelis, Iosif. Exponential deficiency of convolutions of densities. ESAIM: Probability and Statistics, Volume 16 (2012), pp. 86-96. doi : 10.1051/ps/2010010. http://archive.numdam.org/articles/10.1051/ps/2010010/

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