Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-10T23:25:39.901Z Has data issue: false hasContentIssue false
  • English
  • Français

On Exact Sampling of Nonnegative Infinitely Divisible Random Variables

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  04 January 2016

Zhiyi Chi
Zhiyi Chi*
Affiliation:
University of Connecticut
*
∗ Postal address: Department of Statistics, University of Connecticut, 215 Glenbrook Road, U-4120, Storrs, CT 06269, USA. Email address: zhiyi.chi@uconn.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Nonnegative infinitely divisible (i.d.) random variables form an important class of random variables. However, when this type of random variable is specified via Lévy densities that have infinite integrals on (0, ∞), except for some special cases, exact sampling is unknown. We present a method that can sample a rather wide range of such i.d. random variables. A basic result is that, for any nonnegative i.d. random variable X with its Lévy density explicitly specified, if its distribution conditional on X ≤ r can be sampled exactly, where r > 0 is any fixed number, then X can be sampled exactly using rejection sampling, without knowing the explicit expression of the density of X. We show that variations of the result can be used to sample various nonnegative i.d. random variables.

Keywords

Infinitely divisiblerejection samplingexact samplingsubordinatorLévy process

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 842 - 873
Copyright
© Applied Probability Trust 

References

Aalen, O. O. (1992). Modelling heterogeneity in survival analysis by the compound Poisson distribution. Ann. Appl. Prob. 2, 951–972.Google Scholar
Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Nat. Bureau Stand. App. Math. Ser. 55). Government Printing Office, Washington, DC.Google Scholar
Asmussen, S. and Rosiński, J. (2001). Approximations of small Jumps of Lévy processes with a view towards simulation. J. Appl. Prob. 38, 482–493.Google Scholar
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
Bertoin, J. and Yor, M. (2002). The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17, 389–400.Google Scholar
Bondesson, L. (1981). Classes of infinitely divisible distributions and densities. Z. Wahrscheinlichkeitsth. 57, 39–71.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Prob. 31, 929–953.Google Scholar
Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2010). On Lamperti stable processes. Prob. Math. Statist. 30, 1–28.Google Scholar
Chambers, J. M., Mallows, C. L. and Stuck, B. W. (1976). A method for simulating stable random variables. J. Amer. Statist. Assoc. 71, 340–344.Google Scholar
Covo, S. (2009). One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications. Adv. Appl. Prob. 41, 367–392.Google Scholar
Devroye, L. (1986). An automatic method for generating random variates with a given characteristic function. SIAM J. Appl. Math. 46, 698–719.Google Scholar
Devroye, L. (1986). Nonuniform Random Variate Generation. Springer, New York.Google Scholar
Devroye, L. (1987). A simple generator for discrete log-concave distributions. Computing 39, 87–91.Google Scholar
Devroye, L. (2001). Simulating perpetuities. Methodology Comput. Appl. Prob. 3, 97–115.Google Scholar
Devroye, L. and Fawzi, O. (2010). Simulating the Dickman distribution. Statist. Prob. Lett. 80, 242–247.Google Scholar
Fill, J. A. and Huber, M. L. (2010). Perfect simulation of Vervaat perpetuities. Electron. J. Prob. 15, 96–109.Google Scholar
Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering (Appl. Math. 53). Springer, New York.Google Scholar
Hörmann, W., Leydold, J. and Derflinger, G. (2004). Automatic Nonuniform Random Variate Generation. Springer, Berlin.Google Scholar
Hougaard, P. (1986). Survival models for heterogeneous populations derived from stable distributions. Biometrika 73, 387–396.Google Scholar
Kanter, M. (1975). Stable densities under change of scale and total variation inequalities. Ann. Prob. 3, 697–707.Google Scholar
Kendall, W. S. and Thönnes, E. (2004). Random walk CFTP. Tech. Rep., Department of Tech. Rep..Google Scholar
Kolassa, J. E. (1997). Series Approximation Methods in Statistics (Lecture Notes Statist. 88), 2nd edn. Springer, New York.Google Scholar
Kuznetsov, A., Kyprianou, A. E., Pardo, J. and van Schaik, K. (2011). A Wiener-Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Prob. 21, 2171–2190.Google Scholar
Kyprianou, A. E., Pardo, J. C. and Rivero, V. (2010). Exact and asymptotic n-tuple laws at first and last passage. Ann. Appl. Prob. 20, 522–564.Google Scholar
Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrscheinlichkeitsth. 22, 205–225.Google Scholar
Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York.Google Scholar
Papadimitriou, C. H. (1994). Computational Complexity. Addison-Wesley, Reading, MA.Google Scholar
Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.Google Scholar
Reynolds, D. S. and Savage, I. R. (1971). Random wear models in reliability theory. Adv. Appl. Prob. 3, 229–248.Google Scholar
Rudin, W. (1987). Real and Complex Analysis, 3rd edn. McGraw-Hill, New York.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68). Cambridge University Press.Google Scholar
Steutel, F. W. (1970). Preserevation of Infinite Divisibility under Mixing and Related Topics (Math. Centre Tracts 33). Mathematisch Centrum, Amsterdam.Google Scholar
Steutel, F. W. (1979). Infinite divisibility in theory and practice. Scand. J. Statist. 6, 57–64.Google Scholar
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