Convergence of certain classes of random flights in the Kantorovich metric

A random walk of a particle in $\mathbf{R}^d$ is considered. The weak convergence of various transformations of trajectories of random flights with Poisson switching times was studied by Davydov and Konakov in [Random walks in nonhomogeneous Poisson environment, in Modern Problems of Stochastic Analysis and Statistics, Springer, 2017, pp. 3–24], who also built a diffusion approximation of the process of random flights. The goal of the present paper is to prove a stronger convergence with respect to the Kantorovich distance. Three types of transformations are considered. The cases of exponential and superexponential growth of the switching time transformation function are quite simple—in these cases the required result follows from the fact that the limit processes lie within the unit ball. In the case of a power-like growth of the transformation function, the convergence follows from combinatorial arguments and properties of the Kantorovich metric.