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On fixed points of Poisson shot noise transforms

Part of: Special processes Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Aleksander M. Iksanov  and
Zbigniew J. Jurek
Aleksander M. Iksanov*
Affiliation:
Kiev National Taras Shevchenko University
Zbigniew J. Jurek
Affiliation:
University of Wrocław
*
∗ Postal address: Faculty of Cybernetics, Kiev National Taras Shevchenko University, 01033 Kiev, Ukraine. Email address: iksan@unicyb.kiev.ua
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Abstract

Distributional fixed points of a Poisson shot noise transform (for nonnegative and nonincreasing response functions bounded by 1) are characterized. The tail behavior of fixed points is described. Typically they have either exponential moments or their tails are proportional to a power function, with exponent greater than −1. The uniqueness of fixed points is also discussed. Finally, it is proved that in most cases fixed points are absolutely continuous, apart from the possible atom at zero.

Keywords

Shot noise transformfixed pointsregular variationrenewal theoremabsolute continuityinfinite divisibilityBanach contraction principle

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60K05: Renewal theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 798 - 825
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

∗∗

Current address: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.

References

Athreya, K. B. (1969). On the supercritical one dimensional age dependent branching processes. Ann. Math. Statist. 40, 743–763.Google Scholar
Baringhaus, L. and Grübel, R. (1997). On a class of characterization problems for random convex combinations. Ann. Inst. Statist. Math. 49, 555–567.CrossRefGoogle Scholar
Biggins, J. D. and Kyprianou, A. E. (2001). The smoothing transform; the boundary case. Preprint 1214, Department of Mathematics, Utrecht University.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Bondesson, L. (1982). On simulation from infinitely divisible distributions. Adv. Appl. Prob. 14, 855–869.Google Scholar
Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities (Lecture Notes Statist. 76). Springer, New York.Google Scholar
Durrett, R. and Liggett, T. (1983). Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitsth. 64, 275–301.Google Scholar
Embrechts, P. and Goldie, C. M. (1994). Perpetuities and random equations. In Proc. 5th Prague Symp., eds Mandl, P. and Husková, M., Physica, Heidelberg, pp. 75–86.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.Google Scholar
Fill, J. A. and Janson, S. (2000a). A characterization of the set of fixed points of the Quicksort transformation. Electron. Commun. Prob. 5, 77–84.CrossRefGoogle Scholar
Fill, J. A. and Janson, S. (2000b). Smoothness and decay properties of the limiting Quicksort density function. In Mathematics and Computer Science (Proc. Colloq. Math. Comput. Sci., Versailles, 18–20 September 2000), eds Gardy, D. and Mokkadem, A., Birkhäuser, Basel, pp. 53–64.Google Scholar
Fill, J. A. and Janson, S. (2001). Approximating the limiting Quicksort distribution. Random Structures Algorithms 19, 376–406.Google Scholar
Fill, J. A. and Janson, S. (2002). Quicksort asymptotics. J. Algorithms 44, 4–28.Google Scholar
Gikhman, I. I. and Skorokhod, A. V. (1975). Theory of Stochastic Processes, Vol. 2. Springer, Berlin.Google Scholar
Grincevicius, A. K. (1975). One limit distribution for a random walk on the line. Lithuanian Math. J. 15, 580–589.CrossRefGoogle Scholar
Guivarc'h, Y., (1990). Sur une extension de la notion semi-stable. Ann. Inst. Henri Poincaré 26, 261–285.Google Scholar
Iksanov, A. M. (2001). On positive distributions of class L self-decomposable distributions. Teor. Imov. Mat. Statist. 64, 48–56 (in Ukrainian.).Google Scholar
Iksanov, A. M. and Jurek, Z. J. (2003). Shot noise distributions and selfdecomposability. Submitted. To appear in Stoch. Anal. Appl. 21.Google Scholar
Jayakumar, K. and Pillai, R. N. (1996). Characterization of Mittag–Leffler distributons. J. Appl. Statist. Sci. 4, 77–83.Google Scholar
Jurek, Z. J. (1999). Selfdecomposability, stopping times and perpetuity laws. Probab. Math. Statist. 19, 413–419.Google Scholar
Jurek, Z. J. and Mason, J. D. (1993). Operator-limit Distributions in Probability Theory. John Wiley, New York.Google Scholar
Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.Google Scholar
Lau, K. S. and Rao, C. R. (1982). Integrated Cauchy functional equation and characterizations of the exponential law. Sankhyā A 44, 72–90.Google Scholar
Lin, G. D. (2001). A note on the characterization of positive Linnik laws. Austral. N. Z. J. Statist. 43, 17–20.Google Scholar
Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30, 85–112.Google Scholar
Liu, Q. (1999). Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks. Stoch. Process. Appl. 82, 61–87.Google Scholar
Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95, 83–107.Google Scholar
Lukacs, E. (1970). Characteristic Functions, 2nd edn. Griffin, London.Google Scholar
Lyons, R. (1997). A simple path to Biggins's martingale convergence for branching random walk. In Classical and Modern Branching Processes (IMA Vol. Math. Appl. 84), eds Athreya, K. B. and Jagers, P., Springer, Berlin, pp. 217–221.Google Scholar
Rösler, U., (1992). A fixed point theorem for distributions. Stoch. Process. Appl. 42, 195–214.Google Scholar
Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 3–33.CrossRefGoogle Scholar
Rudin, W. (1966). Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York.Google Scholar
Samorodnitsky, G. (1998). Tail behaviour of some shot noise processes. A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, eds Adler, R., Feldman, R. and Taqqu, M. S., Birkhäuser, Boston.Google Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Steutel, F. W. (1970). Preservation of Infinite Divisibility Under Mixing and Related Topics (Math. Centrum Tracts 33). Mathematisch Centrum, Amsterdam.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar
Watanabe, T. (2000). Absolute continuity of some semi-selfdecomposable distributions and self-similar measures. Prob. Theory Relat. Fields, 117, 387–405.Google Scholar