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Subexponential densities of infinitely divisible distributions on the half-line. (English) Zbl 1468.60024

Summary: We show that, under the long-tailedness of the densities of normalized Lévy measures, the densities of infinitely divisible distributions on the half-line are subexponential if and only if the densities of their normalized Lévy measures are subexponential. Moreover, we prove that, under a certain continuity assumption, the densities of infinitely divisible distributions on the half-line are subexponential if and only if their normalized Lévy measures are locally subexponential.

MSC:

60E07 Infinitely divisible distributions; stable distributions
60G51 Processes with independent increments; Lévy processes
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[1] Asmussen, S.; Foss, S.; Korshunov, D., Asymptotics for sums of random variables with local subexponential behaviour, J. Theor. Probab., 16, 2, 489-518 (2003) · Zbl 1033.60053
[2] Borovkov, AA; Borovkov, KA, Asymptotic Analysis of Random Walks (2008), Heavy-Tailed Distributions: Cambridge Univ. Press, Cambridge, Heavy-Tailed Distributions · Zbl 1231.60001
[3] Chistyakov, VP, A theorem on sums of independent positive random variables and its application to branching random processes, Theor. Probab. Appl., 9, 4, 640-648 (1964)
[4] Chover, J.; Ney, P.; Wainger, S., Functions of probability measures, J. Anal. Math., 26, 1, 255-302 (1973) · Zbl 0276.60018
[5] Embrechts, P.; Goldie, CM; Veraverbeke, N., Subexponentiality and infinite divisibility, Z. Wahrscheinlichkeitstheor. Verw. Geb., 49, 3, 335-347 (1979) · Zbl 0397.60024
[6] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling Extremal Events for Insurance and Finance (1997), Berlin: Springer, Berlin · Zbl 0873.62116
[7] Finkelshtein, D.; Tkachov, P., Kesten’s bound for subexponential densities on the real line and its multi-dimensional analogues, Adv. Appl. Probab., 50, 2, 373-395 (2018) · Zbl 1443.60012
[8] Foss, S.; Korshunov, D.; Zachary, S., An Introduction to Heavy-Tailed and SubexponentialDistributions (2013), New York: Springer, New York · Zbl 1274.62005
[9] Foss, S.; Zachary, S., The maximum on a random time interval of a random walk with long-tailed increments and negative drift, Ann. Appl. Probab., 13, 1, 37-53 (2003) · Zbl 1045.60039
[10] Jiang, T.; Wang, Y.; Cui, Z.; Chen, Y., On the almost decrease of a subexponential density, Stat. Probab. Lett., 153, 1, 71-79 (2019) · Zbl 1458.60053
[11] Klüppelberg, C., Subexponential distributions and characterizations of related classes, Probab. Theory Relat. Fields, 82, 2, 259-269 (1989) · Zbl 0687.60017
[12] Korshunov, D., On distribution tail of the maximum of a random walk, Stochastic Processes Appl., 72, 1, 97-103 (1997) · Zbl 0942.60018
[13] Korshunov, D., On the distribution density of the supremum of a random walk in the subexponential case, Sib. Math. J., 47, 6, 1060-1065 (2006) · Zbl 1150.60376
[14] Leslie, JR, On the non-closure under convolution of the subexponential family, J. Appl. Probab., 26, 1, 58-66 (1989) · Zbl 0672.60027
[15] Lin, J., Second order subexponential distributionswith finitemean and their applications to subordinated distributions, J. Theor. Probab., 25, 3, 834-853 (2012) · Zbl 1251.60014
[16] Pakes, AG, Convolution equivalence and infinite divisibility, J. Appl. Probab., 41, 2, 407-424 (2004) · Zbl 1051.60019
[17] Pakes, AG, Convolution equivalence and infinite divisibility: corrections and corollaries, J. Appl. Probab., 44, 2, 295-305 (2007) · Zbl 1132.60015
[18] Pitman, EJG, Subexponential distribution functions, J. Aust. Math. Soc., Ser. A, 29, 3, 337-347 (1980) · Zbl 0425.60012
[19] Rogozin, BA, On the constant in the definition of subexponential distributions, Theor. Probab. Appl., 44, 2, 409-412 (2000) · Zbl 0971.60009 · doi:10.1137/S0040585X97977665
[20] Sato, K., Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ (2013), Cambridge: Press, Cambridge · Zbl 1287.60003
[21] Shimura, T.; Watanabe, T., Infinite divisibility and generalized subexponentiality, Bernoulli, 11, 3, 445-469 (2005) · Zbl 1081.60016
[22] T. Shimura and T. Watanabe, Subexponential densities of compound Poisson sums and the supremum of a random walk, 2020, arXiv:2001.11362.
[23] Teugels, JL, The class of subexponential distributions, Ann. Probab., 3, 6, 1000-1011 (1975) · Zbl 0374.60022
[24] Wang, Y.; Wang, K., Asymptotics of the density of the supremum of a random walk with heavy-tailed increments, J. Appl. Probab., 3, 3, 874-879 (2006) · Zbl 1120.60048
[25] Watanabe, T., Convolution equivalence and distributions of random sums, Probab. Theory Relat. Fields, 142, 3-4, 367-397 (2008) · Zbl 1146.60014
[26] Watanabe, T., Asymptotic properties of Fourier transforms of b-decomposable distributions, J. Fourier Anal. Appl., 18, 4, 803-827 (2012) · Zbl 1257.42018
[27] T. Watanabe, Second order subexponentiality and infinite divisibility, 2020, arXiv:2001.10671. · Zbl 1081.60016
[28] Watanabe, T.; Yamamuro, K., Local subexponentiality of infinitely divisible distributions, J. Math-for-Ind., 1, B, 81-90 (2009) · Zbl 1208.60015
[29] Watanabe, T.; Yamamuro, K., Local subexponentiality and self-decomposability, J. Theor. Probab., 23, 4, 1039-1067 (2010) · Zbl 1215.60014
[30] Watanabe, T.; Yamamuro, K., Tail behaviors of semi-stable distributions, J. Math. Anal. Appl., 393, 1, 108-121 (2012) · Zbl 1250.62026
[31] Watanabe, T.; Yamamuro, K., Two non-closure properties on the class of subexponential densities, J. Theor. Probab., 30, 3, 1059-1075 (2017) · Zbl 1406.60025
[32] Xu, H.; Foss, S.; Wang, Y., On closedness under convolution and convolution roots of the class of long-tailed distributions, Extremes, 18, 4, 605-628 (2015) · Zbl 1327.60042
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