Limit theorems for random walks

TL;DR

This paper studies the convergence of a subordinated random walk to a symmetric alpha-stable process and provides asymptotic formulas for its transition function.

Contribution

It proves the convergence of scaled subordinated random walks to symmetric alpha-stable processes and derives related asymptotic transition formulas.

Findings

Scaled subordinated random walks converge to symmetric alpha-stable processes.

Asymptotic transition function formulas similar to Pólya's formula are established.

Conditions on the subordinator ensure convergence and asymptotic behavior.

Abstract

We consider a random walk $S_{\tau}$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner's subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau}$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha}$. We also prove asymptotic formula for the transition function of $S_{\tau}$ similar to the P\'{o}lya's asymptotic formula for $B^{\alpha}$.