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On Tails of Perpetuities

Part of: Stochastic analysis Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Paweł Hitczenko
Paweł Hitczenko*
Affiliation:
Drexel University
*
∗Postal address: Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA. Email address: phitczenko@math.drexel.edu
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Abstract

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We establish an upper bound on the tails of a random variable that arises as a solution of a stochastic difference equation. In the nonnegative case our bound is similar to a lower bound obtained in Goldie and Grübel (1996).

Keywords

Perpetuitystochastic difference equationtail behavior

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 60H25: Random operators and equations
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1191 - 1194
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Supported in part by the NSA grant #H98230-09-1-0062.

References

[1] Goh, W. M. Y. and Hitczenko, P. (2008). Random partitions with restricted part sizes. Random Structures Algorithms 32, 440–462.Google Scholar
[2] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.CrossRefGoogle Scholar
[3] Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. Appl. Prob. 28, 463–480.Google Scholar
[4] Hitczenko, P. and Wesołowski, J. (2009). Perpetuities with thin tails revisited. Ann. Appl. Prob. 19, 2080–2101. (Corrected (2010) version available at http://arxiv.org/abs/0912.1694v3.)Google Scholar
[5] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.Google Scholar