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 : 10.1051/ps/2010010
Classification : 60E05, 60E10, 60F10, 62E20, 60E15
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},
     year = {2012},
     publisher = {EDP Sciences},
     volume = {16},
     doi = {10.1051/ps/2010010},
     mrnumber = {2946121},
     zbl = {1266.60021},
     language = {en},
     url = {https://www.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

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