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On perpetuities with gamma-like tails

Part of: Distribution theory - Probability Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  26 July 2018

Dariusz Buraczewski ,
Piotr Dyszewski ,
Alexander Iksanov  and
Alexander Marynych
Dariusz Buraczewski*
Affiliation:
University of Wrocław
Piotr Dyszewski*
Affiliation:
University of Wrocław
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv
Alexander Marynych*
Affiliation:
Taras Shevchenko National University of Kyiv
*
* Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
* Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
**** Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine.
**** Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine.
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Abstract

An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We provide three disjoint groups of sufficient conditions which ensure that the right tail of a perpetuity ℙ{X > x} is asymptotic to axce-bx as x → ∞ for some a, b > 0, and c ∈ ℝ. Our results complement those of Denisov and Zwart (2007). As an auxiliary tool we provide criteria for the finiteness of the one-sided exponential moments of perpetuities. We give several examples in which the distributions of perpetuities are explicitly identified.

Keywords

Distribution tailselfdecomposable distributionperpetuityexponential moment

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60G50: Sums of independent random variables; random walks 60E07: Infinitely divisible distributions; stable distributions
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 368 - 389
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Alsmeyer, G. and Dyszewski, P. (2017). Thin tails of fixed points of the nonhomogeneous smoothing transform. Stoch. Process. Appl. 127, 3014–3041. Google Scholar
[2]Alsmeyer, G., Iksanov, A. and Rösler, U. (2009). On distributional properties of perpetuities. J. Theoret. Prob. 22, 666–682. Google Scholar
[3]Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities (Lecture Notes Statist. 76). Springer, New York. Google Scholar
[4]Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331. Google Scholar
[5]Buraczewski, D., Damek, E. and Mikosch, T. (2016). Stochastic Models with Power-Law Tails: The Equation X = AX + B. Springer, Cham. Google Scholar
[6]Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98. Google Scholar
[7]Damek, E. and Dyszewski, P. (2017). Iterated random functions and regularly varying tails. Preprint. Available at https://arxiv.org/abs/1706.03876. Google Scholar
[8]Damek, E. and Kołodziejek, B. (2017). Stochastic recursions: between Kesten's and Grey's assumptions. Preprint. Available at https://arxiv.org/abs/1701.02625. Google Scholar
[9]Denisov, D. and Zwart, B. (2007). On a theorem of Breiman and a class of random difference equations. J. Appl. Prob. 44, 1031–1046. Google Scholar
[10]Dyszewski, P. (2016). Iterated random functions and slowly varying tails. Stoch. Process. Appl. 126, 392–413. Google Scholar
[11]Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166. Google Scholar
[12]Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. Appl. Prob. 28, 463–480. Google Scholar
[13]Goldie, C. M. and Maller, R. A. (2000). Stability of perpetuities. Ann. Prob. 28, 1195–1218. Google Scholar
[14]Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products. Academic Press, San Diego, CA. Google Scholar
[15]Grey, D. R. (1994). Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. Prob. 4, 169–183. Google Scholar
[16]Grincevičius, A. K. (1975). One limit distribution for a random walk on the line. Lithuanian Math. J. 15, 580–589. Google Scholar
[17]Hitczenko, P. (2010). On tails of perpetuities. J. Appl. Prob. 47, 1191–1194. Google Scholar
[18]Hitczenko, P. and Wesołowski, J. (2009). Perpetuities with thin tails revisited. Ann. Appl. Prob. 19, 2080–2101. (Erratum: 20 (2010), 1177.) Google Scholar
[19]Iksanov, A. (2016). Renewal Theory for Perturbed Random Walks and Similar Processes. Birkhäuser, Cham. Google Scholar
[20]Iksanov, A. M. and Jurek, Z. J. (2003). Shot noise distributions and selfdecomposability. Stoch. Anal. Appl. 21, 593–609. Google Scholar
[21]Iksanov, O. M. (2002). On positive distributions of the class L of self-decomposable laws. Theory Prob. Math. Statist. 64, 51–61. Google Scholar
[22]Jacobsen, M., Mikosch, T., Rosiński, J. and Samorodnitsky, G. (2009). Inverse problems for regular variation of linear filters, a cancellation property for σ-finite measures and identification of stable laws. Ann. Appl. Prob. 19, 210–242. Google Scholar
[23]Jurek, Z. J. and Vervaat, W. (1983). An integral representation for self-decomposable Banach space valued random variables. Z. Wahrscheinlichkeitsth. 62, 247–262. Google Scholar
[24]Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248. Google Scholar
[25]Kevei, P. (2016). A note on the Kesten–Grincevičius–Goldie theorem. Electron. Commun. Prob. 21, 51. Google Scholar
[26]Kevei, P. (2017). Implicit renewal theory in the arithmetic case. J. Appl. Prob. 54, 732–749. Google Scholar
[27]Kołodziejek, B. (2017). Logarithmic tails of sums of products of positive random variables bounded by one. Ann. Appl. Prob. 27, 1171–1189. Google Scholar
[28]Konstantinides, D. G., Ng, K. W. and Tang, Q. (2010). The probabilities of absolute ruin in the renewal risk model with constant force of interest. J. Appl. Prob. 47, 323–334. Google Scholar
[29]Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177. Google Scholar
[30]Palmowski, Z. and Zwart, B. (2007). Tail asymptotics of the supremum of a regenerative process. J. Appl. Prob. 44, 349–365. Google Scholar
[31]Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Prob. 14, 612–652. Google Scholar
[32]Takács, L. (1955). On stochastic processes connected with certain physical recording apparatuses. Acta Math. Hungar. 6, 363–380. Google Scholar
[33]Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783. Google Scholar