Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T12:01:14.708Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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