  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.
