Explicit estimates for the asymptotics of subexponential infinitely divisible distribution functions. (English. Russian original) Zbl 0970.60021

Yakymiv, A. L.

Math. Notes 67, No. 2, 239-244 (2000); translation from Mat. Zametki 67, No. 2, 295-301 (2000).

Let \(F(x)\) be an infinitely divisible distribution such that \(F(x)=0\) for \(x<0\), and let \(q(x)\) be the corresponding spectral function. In a special case, when \(q(x)\) has in a sense slow and regular decay, the author obtains order-sharp estimates on the rate of decay of the difference \(|1-F(x)-q(x)|\).

Reviewer: A.Ulanovskij (Khar’kov)

MSC:

60E07	Infinitely divisible distributions; stable distributions
60E15	Inequalities; stochastic orderings

Keywords:

asymptotics of the tails of infinitely divisible distributions

References:

[1]	V. M. Kruglov and C. N. Antonov, ”On the asymptotic behavior of infinitely divisible distributions in Banach space,”Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.],27, No. 4, 625–642 (1982). · Zbl 0503.60026
[2]	V. M. Kruglov and C. N. Antonov, ”Once more on the asymptotic behavior of infinitely divisible distributions in Banach space,”Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.],29, No. 4, 735–742 (1984). · Zbl 0561.60002
[3]	M. S. Sgibnev, ”The asymptotics of infinitely divisible distributions in \(\mathbb{R}\),”Sibirsk. Mat. Zh. [Siberian Math. J.],31, No. 1, 135–140 (1990). · Zbl 0703.60010
[4]	A. L. Yakymiv, ”Asymptotic behavior of of a class of infinitely divisible distributions,”Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.],32, No. 4, 691–702 (1987). · Zbl 0645.60021
[5]	A. L. Yakymiv, ”The asymptotics of the density of an infinitely divisible distribution at infinity,” in:Stability Problems for Stochastic Models [in Russian], VNIISI, Moscow (1990), pp. 123–131.
[6]	P. Embrechts, Ch. M. Goldie, and N. Veraverbeke, ”Subexponentiality and infinite divisibility,”Z. Wahrsch. Verw. Gebiete,49, 333–347 (1979). · Zbl 0397.60024 · doi:10.1007/BF00535504
[7]	R. Grubel, ”Über grenzteilbare Verteilungen,”Arch. Math.,41, No. 1, 80–88 (1983). · Zbl 0504.60018 · doi:10.1007/BF01193826
[8]	E. Omey, ”Infinite divisibility and random sums of random vectors,”Yokohama Math. J.,33, Nos. 1–2, 39–48 (1985). · Zbl 0591.60012
[9]	E. Seneta,Regularly Varying Functions, Springer-Verlag, Berlin-Heidelberg-New York (1976). · Zbl 0324.26002
[10]	V. P. Chistyakov, ”A theorem on the sum of independent positive random variables and its application to branching random processes,”Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.],9, No. 4, 710–718 (1964). · Zbl 0203.19401
[11]	A. L. Yakymiv, ”Sufficient conditions for the subexponential property of the convolution of two distributions,”Mat. Zametki [Math. Notes],58, No. 5, 778–781 (1995). · Zbl 0861.60024
[12]	L. de Haan, ”Regular variation and its application to the weak convergence of sample extremes,”Math. Centre Tracts,32 (1970). · Zbl 0226.60039