Two - sided random walks conditioned to have no intersections ∗

Let S, S be independent simple random walks in Z (d = 2, 3) started at the origin. We construct two-sided random walk paths conditioned that S[0,∞) ∩ S[1,∞) = ∅ by showing the existence of the following limit: lim n→∞ P (· | S[0, τ(n)] ∩ S[1, τ(n)] = ∅), where τ (n) = inf{k ≥ 0 : |S(k)| ≥ n}. Moreover, we give upper bounds of the rate of the convergence. These are discrete analogues of results for Brownian motion obtained in [3] and [8].