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

Abstract

Let ${\mathrm{\xi }}_1$, ${\mathrm{\xi }}_2,\dots$ be i.i.d. random variables of zero mean and finite variance and ${\mathrm{\eta }}_1$, ${\mathrm{\eta }}_2,\dots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an α-stable distribution, $\mathit{\alpha }\in (0,1)$. 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 ${\mathrm{\xi }}_k$ occurs; if the present position of the Markov chain is nonpositive, then the jump ${\mathrm{\eta }}_k$ occurs. We prove functional limit theorems for this and two closely related Markov chains under Donsker’s scaling. The weak limit is a nonnegative process ${(X(t))}_{t\ge 0}$ satisfying a stochastic equation $\mathrm{d}X(t)=\mathrm{d}W(t)+\mathrm{d}{U}_{\mathit{\alpha }}(L_X^{(0)}(t))$, where W is a Brownian motion, $U_{\mathit{\alpha }}$ is an α-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.