Scaling Limits of Random Walk Bridges Conditioned to Avoid a Finite Set

Abstract

running head: Scaling limits of random walk bridges key words: random walk on the integer lattice; conditioned to avoid a set; third moment; functional limit theorem; killing on a finite set; tightness of pinned walk; tunneling.Abstract This paper concerns a scaling limit of a one-dimensional random walk S x n started from x on the integer lattice conditioned to avoid a non-empty finite set A, the random walk being assumed to be irreducible and have zero mean. Suppose the variance σ 2 of the increment law is finite. Given positive constants b, c and T we consider the scaledat another point ≈ −c √ N at t = T and avoid A in between and discuss the functional limit of it as N → ∞. We show that it converges in law to a continuous process if u] to vary regularly as u → −∞ with exponent −β, 2 ≤ β ≤ 3 and show that it converges to a process which has one downward jump that clears the origin if β < 3; in case β = 3 there arises the same limit process as in case E[|S 1 | 3 ; S 1 < 0] < ∞. In case σ 2 = ∞ we consider the special case when S 1 belongs to the domain of attraction of a stable law of index 1 < α < 2 having no negative jumps and obtain analogous results.