Subexponential densities of compound Poisson sums and the supremum of a random walk. (English) Zbl 1512.60010

Shimura, Takaaki; Watanabe, Toshiro

Kyoto J. Math. 63, No. 1, 223-239 (2023).

Based on authors’ abstract: In this paper, the authors study densities of compound Poisson distributions on \((0, \infty)\). Under the condition that Lévy density is square integrable, the authors give several equivalent characterizations for the density of the compound Poisson distribution to be subexponential, and show that the class of probability density functions on \(\mathbb{R}_+\) is closed under generalized convolutions roots for compound Poisson sums. As an application, the authors characterize the subexponentiality of the density on \((0, \infty)\) of the distribution of the supremum of a random walk.

Reviewer: Renming Song (Urbana)

MSC:

60E07	Infinitely divisible distributions; stable distributions
60G50	Sums of independent random variables; random walks

Keywords:

compound Poisson sum; local subexponentiality; random walk; subexponential density

References:

[1]	S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd ed., Adv. Ser. Stat. Sci. Appl. Probab. 14, World Scientific Publishing, Hackensack, 2010. · Zbl 1247.91080 · doi:10.1142/9789814282536
[2]	S. Asmussen, S. Foss, and, D. Korshunov, Asymptotics for sums of random variables with local subexponential behaviour, J. Theoret. Probab. 16 (2003), no. 2, 489-518. · Zbl 1033.60053 · doi:10.1023/A:1023535030388
[3]	S. Asmussen, V. Kalashnikov, D. Konstantinides, C. Klüppelberg, and G. Tsitsiashvili, A local limit theorem for random walk maxima with heavy tails, Statist. Probab. Lett. 56 (2002), no. 4, 399-404. · Zbl 0997.60047 · doi:10.1016/S0167-7152(02)00033-0
[4]	J. Chover, P. Ney, and S. Wainger, Functions of probability measures, J. Anal. Math. 26 (1973), 255-302. · Zbl 0276.60018 · doi:10.1007/BF02790433
[5]	D. Finkelshtein and P. Tkachov, Kesten’s bound for subexponential densities on the real line and its multi-dimensional analogues, Adv. Appl. Probab. 50 (2018), no. 2, 373-395. · Zbl 1443.60012 · doi:10.1017/apr.2018.18
[6]	S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd ed., Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 2013. · Zbl 1274.62005 · doi:10.1007/978-1-4614-7101-1
[7]	S. Foss and S. Zachary, The maximum on a random time interval of a random walk with long-tailed increments and negative drift, Ann. Appl. Probab. 13 (2003), no. 1, 37-53. · Zbl 1045.60039 · doi:10.1214/aoap/1042765662
[8]	C. Klüppelberg, Subexponential distributions and integrated tails, J. Appl. Probab. 25 (1988), no. 1, 132-141. · Zbl 0651.60020 · doi:10.2307/3214240
[9]	C, Subexponential distributions and characterizations of related classes, Probab. Theory Related Fields 82 (1989), no. 2, 259-269. · Zbl 0687.60017 · doi:10.1007/BF00354763
[10]	D. Korshunov, On the distribution density of the supremum of a random walk in the subexponential case, Siberian Math. J. 47 (2006), no. 6, 1060-1065. · Zbl 1150.60376 · doi:10.1007/s11202-006-0115-5
[11]	K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Stud. Adv. Math. 68, Cambridge Univ. Press, Cambridge, 2013. · Zbl 1287.60003
[12]	T. Watanabe, Convolution equivalence and distributions of random sums, Probab. Theory Related Fields 142 (2008), nos. 3-4, 367-397. · Zbl 1146.60014 · doi:10.1007/s00440-007-0109-7
[13]	T, The Wiener condition and the conjectures of Embrechts and Goldie, Ann. Probab. 47 (2019), no. 3, 1221-1239. · Zbl 1466.60026 · doi:10.1214/17-AOP1195
[14]	T, Two hypotheses on the exponential class in the class of O-subexponential infinitely divisible distributions, J. Theoret. Probab. 34 (2021), no. 2, 852-873. · Zbl 1479.60036 · doi:10.1007/s10959-019-00976-z
[15]	T, Second order subexponentiality and infinite divisibility, J. Aust. Math. Soc. 112 (2022), no. 3, 367-390. · Zbl 1487.60034 · doi:10.1017/S1446788720000208
[16]	T, Subexponential densities of infinitely divisible distributions on the half-line, Lithuanian Math. J. 60 (2020), no. 4, 530-543. · Zbl 1468.60024 · doi:10.1007/s10986-020-09495-5
[17]	T. Watanabe and K. Yamamuro, Local subexponentiality and self-decomposability, J. Theoret. Probab. 23 (2010), no. 4, 1039-1067. · Zbl 1215.60014 · doi:10.1007/s10959-009-0240-8
[18]	T,Two non-closure properties on the class of subexponential densities, J. Theoret. Probab. 30 (2017), no. 3, 1059-1075. · Zbl 1406.60025 · doi:10.1007/s10959-016-0672-x