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

Abstract

Let ξ1, ξ2,… be i.i.d. random variables of zero mean and finite variance and η1, η2,… positive i.i.d. random variables whose distribution belongs to the domain of attraction of an α-stable distribution, α∈(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 ξk occurs; if the present position of the Markov chain is nonpositive, then the jump ηk 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≥0 satisfying a stochastic equation dX(t)=dW(t)+dUα(LX(0)(t)), where W is a Brownian motion, Uα is an α-stable subordinator which is independent of W, and LX(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.