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Fourier transform MCMC, heavy tailed distributions and geometric ergodicity

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  • Denis Belomestny
  • Leonid Iosipoi

Abstract

Markov Chain Monte Carlo methods become increasingly popular in applied mathematics as a tool for numerical integration with respect to complex and high-dimensional distributions. However, application of MCMC methods to heavy tailed distributions and distributions with analytically intractable densities turns out to be rather problematic. In this paper, we propose a novel approach towards the use of MCMC algorithms for distributions with analytically known Fourier transforms and, in particular, heavy tailed distributions. The main idea of the proposed approach is to use MCMC methods in Fourier domain to sample from a density proportional to the absolute value of the underlying characteristic function. A subsequent application of the Parseval's formula leads to an efficient algorithm for the computation of integrals with respect to the underlying density. We show that the resulting Markov chain in Fourier domain may be geometrically ergodic even in the case of heavy tailed original distributions. We illustrate our approach by several numerical examples including multivariate elliptically contoured stable distributions.

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  • Denis Belomestny & Leonid Iosipoi, 2019. "Fourier transform MCMC, heavy tailed distributions and geometric ergodicity," Papers 1909.00698, arXiv.org, revised Dec 2019.
    • Handle: RePEc:arx:papers:1909.00698
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    References listed on IDEAS

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    1. Paul Glasserman & Zongjian Liu, 2010. "Sensitivity Estimates from Characteristic Functions," Operations Research, INFORMS, vol. 58(6), pages 1611-1623, December.
    2. Jarner, Søren Fiig & Hansen, Ernst, 2000. "Geometric ergodicity of Metropolis algorithms," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 341-361, February.
    3. Arnak S. Dalalyan, 2017. "Theoretical guarantees for approximate sampling from smooth and log-concave densities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 651-676, June.
    4. John Nolan, 2013. "Multivariate elliptically contoured stable distributions: theory and estimation," Computational Statistics, Springer, vol. 28(5), pages 2067-2089, October.
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