Asymptotic behaviour of non-isotropic random walks with heavy tails

TL;DR

This paper investigates the asymptotic behavior of non-isotropic random walks with heavy-tailed flight lengths, deriving Gaussian and Cauchy-based limit theorems under different distributional assumptions.

Contribution

It provides new limit theorems for non-isotropic random walks with exponential and heavy-tailed flight lengths, expanding understanding of their asymptotic distributions.

Findings

Gaussian limit for exponential flight lengths

Convolution of Cauchy and Gaussian distributions for heavy-tailed flight lengths

Asymptotic behavior depends on the tail distribution of flight lengths

Abstract

A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.