Exponential deficiency of convolutions of densities
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 86-96.

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 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.

DOI : https://doi.org/10.1051/ps/2010010
Classification : 60E05,  60E10,  60F10,  62E20,  60E15
Mots clés : 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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     author = {Pinelis, Iosif},
     title = {Exponential deficiency of convolutions of densities},
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     pages = {86--96},
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     mrnumber = {2946121},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2010010/}
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Pinelis, Iosif. Exponential deficiency of convolutions of densities. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 86-96. doi : 10.1051/ps/2010010. http://archive.numdam.org/articles/10.1051/ps/2010010/

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