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.

中文翻译：

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傅里叶变换 MCMC、重尾分布和几何遍历性

马尔可夫链蒙特卡罗方法在应用数学中越来越流行，作为对复杂和高维分布进行数值积分的工具。然而，将 MCMC 方法应用于重尾分布和具有难以分析的密度的分布被证明是相当有问题的。在本文中，我们提出了一种新方法，将 MCMC 算法用于具有解析已知傅立叶变换的分布，特别是重尾分布。所提出方法的主要思想是在傅立叶域中使用 MCMC 方法从与基础特征函数的绝对值成正比的密度中进行采样。Parseval 的后续应用 s 公式导致了一种有效的算法，用于计算关于基础密度的积分。我们表明，即使在重尾原始分布的情况下，傅立叶域中的结果马尔可夫链也可能是几何遍历的。我们通过几个数值例子来说明我们的方法，包括多元椭圆轮廓稳定分布。