Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T12:59:22.286Z Has data issue: false hasContentIssue false
  • English
  • Français

On perpetuities with light tails

Part of: Stochastic analysis Distribution theory - Probability

Published online by Cambridge University Press:  29 November 2018

Bartosz Kołodziejek
Bartosz Kołodziejek*
Affiliation:
Warsaw University of Technology
*
* Postal address: Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the asymptotics of logarithmic tails of a perpetuity R=D∑j=1∞Qj∏k=1j-1Mk, where (Mn,Qn)n=1∞ are independent and identically distributed copies of (M,Q), for the case when ℙ(M∈[0,1))=1 and Q has all exponential moments. If M and Q are independent, under regular variation assumptions, we find the precise asymptotics of -logℙ(R>x) as x→∞. Moreover, we deal with the case of dependent M and Q, and give asymptotic bounds for -logℙ(R>x). It turns out that the dependence structure between M and Q has a significant impact on the asymptotic rate of logarithmic tails of R. Such a phenomenon is not observed in the case of heavy-tailed perpetuities.

Keywords

Perpetuitydependence structureregular variationTauberian theoremconvex conjugate

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60E99: None of the above, but in this section
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1119 - 1154
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.,Iksanov, A. and Rösler, U. (2009).On distributional properties of perpetuities.J. Theoret. Prob. 22,666–682.Google Scholar
[2]Bingham, N. H.,Goldie, C. M. and Teugels, J. L. (1989).Regular Variation (Encyclopedia Math. Appl. 27).Cambridge University Press.Google Scholar
[3]Buraczewski, D.,Damek, E. and Mikosch, T. (2016).Stochastic Models with Power-Law Tails. The Equation X=AX+B.Springer,Cham.Google Scholar
[4]Buraczewski, D.,Dyszewski, P.,Iksanov, A. and Marynych, A. (2018).On perpetuities with gamma-like tails.J. Appl. Prob. 55,368–389.Google Scholar
[5]Cambanis, S.,Simons, G. and Stout, W. (1976).Inequalities for Ek(X,Y) when the marginals are fixed.Z Wahrscheinlichkeitsth. 36,285–294.Google Scholar
[6]Daley, D. J. and Goldie, C. M. (2006).The moment index of minima. II.Statist. Prob. Lett. 76,831–837.Google Scholar
[7]Damek, E. and Kołodziejek, B. (2017). Stochastic recursions: between Kesten's and Grey's assumptions. Preprint. Available at https://arxiv.org/abs/1701.02625v1.Google Scholar
[8]Diaconis, P. and Freedman, D. (1999).Iterated random functions.SIAM Rev. 41,45–76.Google Scholar
[9]Dyszewski, P. (2016).Iterated random functions and slowly varying tails.Stoch. Process. Appl. 126,392–413.Google Scholar
[10]Goldie, C. M. (1991).Implicit renewal theory and tails of solutions of random equations.Ann. Appl. Prob. 1,126–166.Google Scholar
[11]Goldie, C. M. and Grübel, R. (1996).Perpetuities with thin tails.Adv. Appl. Prob. 28,463–480.Google Scholar
[12]Goldie, C. M. and Maller, R. A. (2000).Stability of perpetuities.Ann. Prob. 28,1195–1218.Google Scholar
[13]Grey, D. R. (1994).Regular variation in the tail behaviour of solutions of random difference equations.Ann. Appl. Prob. 4,169–183.Google Scholar
[14]Hiriart-Urruty, J.-B. and Lemaréchal, C. (1993).Convex Analysis and Minimization Algorithms. II (Fundamental Principles Math. Sci. 306).Springer,Berlin.Google Scholar
[15]Hitczenko, P. (2010).On tails of perpetuities.J. Appl. Prob. 47,1191–1194.Google Scholar
[16]Hitczenko, P. and Wesołowski, J. (2009).Perpetuities with thin tails revisited.Ann. Appl. Prob. 19,2080–2101.Google Scholar
[17]Iksanov, A. (2016).Renewal Theory for Perturbed Random Walks and Similar Processes.Birkhäuser/Springer,Cham.Google Scholar
[18]Iksanov, A. and Möhle, M. (2008).On the number of jumps of random walks with a barrier.Adv. Appl. Prob. 40,206–228.Google Scholar
[19]Kevei, P. (2016).A note on the Kesten-Grincevičius-Goldie theorem.Electron. Commun. Prob. 21, 12pp.Google Scholar
[20]Kimeldorf, G. and Sampson, A. R. (1989).A framework for positive dependence.Ann. Inst. Statist. Math. 41,31–45.Google Scholar
[21]Kołodziejek, B. (2017).Logarithmic tails of sums of products of positive random variables bounded by one.Ann. Appl. Prob. 27,1171–1189.Google Scholar
[22]Lehmann, E. L. (1966).Some concepts of dependence.Ann. Math. Statist. 37,1137–1153.Google Scholar
[23]Lorentz, G. G. (1953).An inequality for rearrangements.Amer. Math. Monthly 60,176–179.Google Scholar
[24]Maulik, K.,Resnick, S. and Rootzén, H. (2002).Asymptotic independence and a network traffic model.J Appl. Prob. 39,671–699.Google Scholar
[25]Rivero, V. (2003).A law of iterated logarithm for increasing self-similar Markov processes.Stoch. Stoch. Rep. 75,443–472.Google Scholar
[26]Rockafellar, R. T. (1970).Convex Analysis (Princeton Math. Ser. 28).Princeton University Press.Google Scholar
[27]Rothschild, M. and Stiglitz, J. E. (1970).Increasing risk. I. A definition.J. Econom. Theory 2,225–243.Google Scholar
[28]Tang, Q. and Yuan, Z. (2016).Random difference equations with subexponential innovations.Sci. China Math. 59,2411–2426.Google Scholar
[29]Tchen, A. H. (1980).Inequalities for distributions with given marginals.Ann. Prob. 8,814–827.Google Scholar
[30]Vervaat, W. (1972).Success Epochs in Bernoulli Trials (with Applications in Number Theory) (Math. Centre Tracts 42).Mathematisch Centrum,Amsterdam.Google Scholar
[31]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