Functional limit theorems for random walks perturbed by positive alpha-stable jumps

TL;DR

This paper establishes a functional limit theorem for a Markov chain with mixed Gaussian and heavy-tailed jumps, converging to a process combining Brownian motion and an alpha-stable subordinator, revealing complex boundary behavior.

Contribution

It introduces a new limit theorem for Markov chains with positive alpha-stable jumps, characterizing the weak limit as a process with a stochastic differential equation involving local time.

Findings

Weak convergence to a process satisfying a stochastic equation

The limit process is a Feller Brownian motion with jump-type exit at zero

The process combines Brownian motion and an alpha-stable subordinator

Abstract

Let $\xi_1$, $\xi_2,\ldots$ be i.i.d. random variables of zero mean and finite variance and $\eta_1$, $\eta_2,\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $\alpha$-stable distribution, $\alpha\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $\xi_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $\eta_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. The weak limit is a nonnegative process $(X(t))_{t\geq 0}$ satisfying a stochastic equation ${\rm d}X(t)={\rm d}W(t)+ {\rm d}U_\alpha(L_X^{(0)}(t))$, where $W$ is a Brownian motion, $U_\alpha$ is an $\alpha$-stable subordinator which is independent of $W$, and $L_X^{(0)}$ is a local time of $X$ at $0$. Also, we explain that $X$ is a Feller Brownian motion with a `jump-type' exit from $0$.