Journal of Applied Probability

Markov chain Monte Carlo for computing rare-event probabilities for a heavy-tailed random walk

Thorbjörn Gudmundsson and Henrik Hult

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Abstract

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

Article information

Source
J. Appl. Probab., Volume 51, Number 2 (2014), 359-376.

Dates
First available in Project Euclid: 12 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1402578630

Digital Object Identifier
doi:10.1239/jap/1402578630

Mathematical Reviews number (MathSciNet)
MR3217772

Zentralblatt MATH identifier
1291.65014

Subjects
Primary: 65C05: Monte Carlo methods 60J22: Computational methods in Markov chains [See also 65C40]
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
Markov chain Monte Carlo heavy tail rare-event simulation random walk

Citation

Gudmundsson, Thorbjörn; Hult, Henrik. Markov chain Monte Carlo for computing rare-event probabilities for a heavy-tailed random walk. J. Appl. Probab. 51 (2014), no. 2, 359--376. doi:10.1239/jap/1402578630. https://projecteuclid.org/euclid.jap/1402578630


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