Fourier transform MCMC, heavy tailed distributions and geometric ergodicity

TL;DR

This paper introduces a Fourier domain MCMC approach for heavy tailed distributions, enabling efficient integration and demonstrating geometric ergodicity even for complex distributions.

Contribution

It proposes a novel Fourier domain MCMC method for heavy tailed distributions with known characteristic functions, ensuring geometric ergodicity.

Findings

Efficient integral computation using Fourier MCMC.

Applicable to heavy tailed distributions like stable laws.

Demonstrated geometric ergodicity in numerical examples.

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.