Convergence of some classes of random flights in Wasserstein distance

TL;DR

This paper proves stronger convergence of certain random flight processes in Wasserstein distance, extending previous results by analyzing different growth cases of switching moments and employing combinatorial methods.

Contribution

It introduces new convergence results in Wasserstein distance for random flights with Poisson switching, including cases of exponential, super-exponential, and power growth.

Findings

Convergence in Wasserstein distance is established for exponential and super-exponential growth cases.

The power growth case requires complex combinatorial estimation techniques.

Results extend previous convergence analyses of random flight processes.

Abstract

In this paper we consider a random walk of a particle in $\mathbb{R}^d$. Convergence of different transformations of trajectories of random flights with Poisson switching moments has been obtained by Davydov and Konakov, as well as diffusion approximation of the process has been built. The goal of this paper is to prove stronger convergence in terms of the Wasserstein distance. Three types of transformations are considered: cases of exponential and super-exponential growth of a switching moment transformation function are quite simple, and the result follows from the fact that the limit processes belong to the unit ball. In the case of the power growth the estimation is more complicated and follows from combinatorial reasoning and properties of the Wasserstein metric.