Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T19:48:32.798Z Has data issue: false hasContentIssue false
  • English
  • Français

Importance sampling of heavy-tailed iterated random functions

Part of: Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  16 November 2018

Bohan Chen ,
Chang-Han Rhee  and
Bert Zwart
Bohan Chen*
Affiliation:
Centrum Wiskunde & Informatica
Chang-Han Rhee*
Affiliation:
Centrum Wiskunde & Informatica
Bert Zwart*
Affiliation:
Centrum Wiskunde & Informatica
*
* Postal address: Stochastics Group, Centrum Wiskunde & Informatica, Science Park 123, 1098 XG, Amsterdam, The Netherlands.
*** Current address: Industrial Engineering and Management Sciences, 2145 Sheridan Road, Tech C150, Evanston, Illinois, IL 60208-3109, USA. Email address: [email protected]
* Postal address: Stochastics Group, Centrum Wiskunde & Informatica, Science Park 123, 1098 XG, Amsterdam, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the stationary solution Z of the Markov chain {Zn}n∈ℕ defined by Zn+1=ψn+1(Zn), where {ψn}n∈ℕ is a sequence of independent and identically distributed random Lipschitz functions. We estimate the probability of the event {Z>x} when x is large, and develop a state-dependent importance sampling estimator under a set of assumptions on ψn such that, for large x, the event {Z>x} is governed by a single large jump. Under natural conditions, we show that our estimator is strongly efficient. Special attention is paid to a class of perpetuities with heavy tails.

Keywords

State-dependent importance samplingheavy-tailed distributioniterated random functionperpetuities

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 805 - 832
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]Asmussen, S. (2003). Applied Probability and Queues. Springer, New York.Google Scholar
[2]Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.Google Scholar
[3]Asmussen, S. and Nielsen, H. M. (1995). Ruin probabilities via local adjustment coefficients. J. Appl. Prob. 32, 736–755.Google Scholar
[4]Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probabilities for non-standard risk and queueing processes with subexponential jumps. Adv. Appl. Prob. 31, 422–447.Google Scholar
[5]Blanchet, J. and Glynn, P. (2008). Efficient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Prob. 18, 1351–1378.Google Scholar
[6]Blanchet, J., Lam, H. and Zwart, B. (2012). Efficient rare-event simulation for perpetuities. Stoch. Process. Appl. 122, 3361–3392.Google Scholar
[7]Blanchet, J. and Zwart, B. (2007). Importance sampling of compounding processes. In Proc. 2007 Winter Simulation Conference, IEEE, pp. 372–379.Google Scholar
[8]Buraczewski, D., Damek, E. and Mikosch, T. (2016). Stochastic Models with Power-Law Tails: The Equation X=AX+B. Springer, Cham.Google Scholar
[9]Collamore, J. F. and Vidyashankar, A. N. (2013). Tail estimates for stochastic fixed point equations via nonlinear renewal theory. Stoch. Process. Appl. 123, 3378–3429.Google Scholar
[10]Collamore, J. F., Diao, G. and Vidyashankar, A. N. (2014). Rare event simulation for processes generated via stochastic fixed point equations. Ann. Appl. Prob. 24, 2143–2175.Google Scholar
[11]Douc, R., Moulines, E. and Soulier, P. (2007). Computable convergence rates for sub-geometric ergodic Markov chains. Bernoulli 13, 831–848.Google Scholar
[12]Dyszewski, P. (2016). Iterated random functions and slowly varying tails. Stoch. Process. Appl. 126, 392–413.Google Scholar
[13]Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Springer, Berlin.Google Scholar
[14]Foss, S., Korshunov, D. and Zachary, S. (2013). An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn. Springer, New York.Google Scholar
[15]Ganesh, A., O'Connell, N. and Wischik, D. (2004). Big Queues. Springer, Berlin.Google Scholar
[16]Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.Google Scholar
[17]Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. Appl. Prob. 28, 463–480.Google Scholar
[18]Grey, D. R. (1994). Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. Prob. 4, 169–183.Google Scholar
[19]Grincevičius, A. K. (1975). Limit theorems for products of random linear transformations on the line. Lithuanian Math. J. 15, 568–579.Google Scholar
[20]Jarner, S. F. and Roberts, G. O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Prob. 12, 224–247.Google Scholar
[21]Kalashnikov, V. and Tsitsiashvili, G. (1999). Tails of waiting times and their bounds. Queueing Systems Theory Appl. 32, 257–283.Google Scholar
[22]Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.Google Scholar
[23]Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
[24]Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177.Google Scholar
[25]Mirek, M. (2011). Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Prob. Theory Relat. Fields 151, 705–734.Google Scholar
[26]Pakes, A. G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555–564.Google Scholar
[27]Palmowski, Z. and Zwart, B. (2007). Tail asymptotics of the supremum of a regenerative process. J. Appl. Prob. 44, 349–365.Google Scholar
[28]Rhee, C.-H. and Glynn, P. W. (2015). Unbiased estimation with square root convergence for SDE models. Operat. Res. 63, 1026–1043.Google Scholar
[29]Veraverbeke, N. (1977). Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stoch. Process. Appl. 5, 27–37.Google Scholar
[30]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
[31]Zachary, S. (2004). A note on Veraverbeke's theorem. Queueing Systems 46, 9–14.Google Scholar