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 ${\stackrel{˜}{p}}_{t}$˜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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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/

[1] O. Barndorff-Nielsen and D.R. Cox, Edgeworth and saddle-point approximations with statistical applications. J. R. Stat. Soc., Ser. B 41 (1979) 279-312. With discussion. | MR 557595 | Zbl 0424.62010

[2] R.N. Bhattacharya and R.R. Rao, Normal approximation and asymptotic expansions. Robert E. Krieger Publishing Co. Inc., Melbourne, FL (1986). Reprint of the 1976 original. | MR 855460 | Zbl 0331.41023

[3] H.E. Daniels, Tail probability approximations. Int. Stat. Rev. 55 (1987) 37-48. | MR 962940 | Zbl 0614.62016

[4] P. Embrechts and C.M. Goldie, On convolution tails. Stoch. Proc. Appl. 13 (1982) 263-278. | MR 671036 | Zbl 0487.60016

[5] B.-Y. Jing, Q.-M. Shao and W. Zhou, Saddlepoint approximation for Student's t-statistic with no moment conditions. Ann. Stat. 32 (2004) 2679-2711. | MR 2153999 | Zbl 1068.62016

[6] C. Klüppelberg, Subexponential distributions and characterizations of related classes. Probab. Theory Relat. Fields 82 (1989) 259-269. | MR 998934 | Zbl 0687.60017

[7] R. Lugannani and S. Rice, Saddle point approximation for the distribution of the sum of independent random variables. Adv. Appl. Probab. 12 (1980) 475-490. | MR 569438 | Zbl 0425.60042

[8] I.F. Pinelis, Asymptotic equivalence of the probabilities of large deviations for sums and maximum of independent random variables, in Limit theorems of probability theory. “Nauka” Sibirsk. Otdel., Novosibirsk. Trudy Inst. Mat. 5 (1985) 144-173, 176. | MR 821760 | Zbl 0607.60023

[9] N. Reid, Saddlepoint methods and statistical inference. Stat. Sci. 3 (1988) 213-238. With comments and a rejoinder by the author. | MR 968390 | Zbl 0955.62541

[10] Q.-M. Shao, Recent progress on self-normalized limit theorems, in Probability, finance and insurance. World Sci. Publ., River Edge, NJ (2004) 50-68. | MR 2189198

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