On subordinate random walks

TL;DR

This paper investigates the subordination of random walks in Euclidean space, establishing equivalence with Lévy process subordination and characterizing convergence to stable processes based on the subordinator’s Laplace exponent.

Contribution

It proves the equivalence of subordination of random walks and Lévy processes and characterizes the convergence to stable processes via the regular variation of the Laplace exponent.

Findings

Subordination of random walks matches subordination of Lévy processes.

Scaled subordinate random walks converge to stable processes under specific conditions.

Convergence depends on the regular variation of the Laplace exponent at zero.

Abstract

In this article subordination of random walks in $R^d$ is considered. We prove that subordination of random walks in the sense of [BSC12] yields the same process as subordination of L\'evy processes (in the sense of Bochner). Furthermore, we prove that appropriately scaled subordinate random walk converges to a multiple of a rotationally $2\alpha$-stable process if and only if the Laplace exponent of the corresponding subordinator varies regularly at zero with index $\alpha\in (0,1]$.