×

zbMATH — the first resource for mathematics

On exact sampling of nonnegative infinitely divisible random variables. (English) Zbl 1261.60023
The author explains how a rather wide range of nonnegative infinitely divisible random variables, which are specified by Lévy measures, can be sampled exactly. The main result can be described as follows: for any nonnegative infinitely divisible random variable \(X\) of a specified Lévy measure, if its distribution conditional on \(X\leq r\) 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 \(X\). The suggested procedure does not rely on explicit approximations to distributions, complements other available sampling procedures and, mainly, enables exact sampling of nonnegative infinitely divisible random variables with Lévy measures having infinite integrals on \((0,\infty)\).

MSC:
60E07 Infinitely divisible distributions; stable distributions
62D05 Sampling theory, sample surveys
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Aalen, O. O. (1992). Modelling heterogeneity in survival analysis by the compound Poisson distribution. Ann. Appl. Prob. 2 , 951-972. · Zbl 0762.62031
[2] 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. · Zbl 0171.38503
[3] Asmussen, S. and Rosiński, J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Prob. 38 , 482-493. · Zbl 0989.60047
[4] Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121 ). Cambridge University Press. · Zbl 0861.60003
[5] 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. · Zbl 1004.60046
[6] Bondesson, L. (1981). Classes of infinitely divisible distributions and densities. Z. Wahrscheinlichkeitsth. 57 , 39-71. · Zbl 0464.60016
[7] Breiman, L. (1968). Probability . Addison-Wesley, Reading, MA. · Zbl 0174.48801
[8] Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Prob. 31 , 929-953. · Zbl 0957.60055
[9] Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2010). On Lamperti stable processes. Prob. Math. Statist. 30 , 1-28. · Zbl 1198.60022
[10] 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. · Zbl 0341.65003
[11] Covo, S. (2009). One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications. Adv. Appl. Prob. 41 , 367-392. · Zbl 1169.60316
[12] Devroye, L. (1986). An automatic method for generating random variates with a given characteristic function. SIAM J. Appl. Math. 46 , 698-719. · Zbl 0615.65002
[13] Devroye, L. (1986). Nonuniform Random Variate Generation . Springer, New York. · Zbl 0593.65005
[14] Devroye, L. (1987). A simple generator for discrete log-concave distributions. Computing 39 , 87-91. · Zbl 0626.65002
[15] Devroye, L. (2001). Simulating perpetuities. Methodology Comput. Appl. Prob. 3 , 97-115. · Zbl 0982.65005
[16] Devroye, L. and Fawzi, O. (2010). Simulating the Dickman distribution. Statist. Prob. Lett. 80 , 242-247. · Zbl 1180.62028
[17] Fill, J. A. and Huber, M. L. (2010). Perfect simulation of Vervaat perpetuities. Electron. J. Prob. 15 , 96-109. · Zbl 1202.60120
[18] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering (Appl. Math. 53 ). Springer, New York. · Zbl 1038.91045
[19] Hörmann, W., Leydold, J. and Derflinger, G. (2004). Automatic Nonuniform Random Variate Generation . Springer, Berlin. · Zbl 1038.65002
[20] Hougaard, P. (1986). Survival models for heterogeneous populations derived from stable distributions. Biometrika 73 , 387-396. · Zbl 0603.62015
[21] Kanter, M. (1975). Stable densities under change of scale and total variation inequalities. Ann. Prob. 3 , 697-707. · Zbl 0323.60013
[22] Kendall, W. S. and Thönnes, E. (2004). Random walk CFTP. Tech. Rep., Department of Statistics, University of Warwick.
[23] Kolassa, J. E. (1997). Series Approximation Methods in Statistics (Lecture Notes Statist. 88 ), 2nd edn. Springer, New York. · Zbl 0877.62013
[24] 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. · Zbl 1245.65005
[25] 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. · Zbl 1200.60038
[26] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrscheinlichkeitsth. 22 , 205-225. · Zbl 0274.60052
[27] Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing . Springer, New York. · Zbl 0991.65001
[28] Papadimitriou, C. H. (1994). Computational Complexity . Addison-Wesley, Reading, MA. · Zbl 0833.68049
[29] 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. · Zbl 0859.60067
[30] Reynolds, D. S. and Savage, I. R. (1971). Random wear models in reliability theory. Adv. Appl. Prob. 3 , 229-248. · Zbl 0235.60088
[31] Rudin, W. (1987). Real and Complex Analysis , 3rd edn. McGraw-Hill, New York. · Zbl 0925.00005
[32] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes . Chapman & Hall, New York. · Zbl 0925.60027
[33] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68 ). Cambridge University Press. · Zbl 0973.60001
[34] Steutel, F. W. (1970). Preserevation of Infinite Divisibility under Mixing and Related Topics (Math. Centre Tracts 33 ). Mathematisch Centrum, Amsterdam. · Zbl 0226.60013
[35] Steutel, F. W. (1979). Infinite divisibility in theory and practice. Scand. J. Statist. 6 , 57-64. · Zbl 0402.62007
[36] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11 , 750-783. · Zbl 0417.60073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.