Theory of Probability &Its Applications, Volume 65, Issue 4, Page 656-664, January 2021.
A random walk of a particle in ${R}^d$ 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.

中文翻译：

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Kantorovich 度量中某些类别的随机飞行的收敛性

概率论及其应用，第 65 卷，第 4 期，第 656-664 页，2021 年 1 月。
考虑粒子在 ${R}^d$ 中的随机游走。Davydov 和 Konakov 在 [Random walks in nonhomogeneous Poisson environment, in Modern Problems of Stochastic Analysis and Statistics, Springer, 2017, pp. 3--24] 中研究了具有泊松切换时间的随机飞行轨迹的各种变换的弱收敛性，他还建立了随机飞行过程的扩散近似。本文的目标是证明关于 Kantorovich 距离的更强收敛性。考虑了三种类型的转换。开关时间变换函数的指数和超指数增长的情况非常简单——在这些情况下，所需的结果来自极限过程位于单位球内的事实。