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Markov Chain Monte Carlo methods become increasingly popular in appliedmathematics as a tool for numerical integration with respect to complex andhigh-dimensional distributions. However, application of MCMC methods to heavytailed distributions and distributions with analytically intractable densitiesturns out to be rather problematic. In this paper, we propose a novel approachtowards the use of MCMC algorithms for distributions with analytically knownFourier transforms and, in particular, heavy tailed distributions. The mainidea of the proposed approach is to use MCMC methods in Fourier domain tosample from a density proportional to the absolute value of the underlyingcharacteristic function. A subsequent application of the Parseval's formulaleads to an efficient algorithm for the computation of integrals with respectto the underlying density. We show that the resulting Markov chain in Fourierdomain may be geometrically ergodic even in the case of heavy tailed originaldistributions. We illustrate our approach by several numerical examplesincluding multivariate elliptically contoured stable distributions.

Fourier transform MCMC, heavy tailed distributions and geometric ergodicity