Escape time for a random walk from an orthant

AbstractLet {(ξk, ηk), k>⩾} be a sequence of independent random vectors with values in {-1, 0, …} ×{−1, 0, …}. Assume the component variables have zero means, bounded second moments, and that α = E[ξkηk] is the same for all k. Let Zn denote (i0,j0)+Σn1 (ξk, where i0, j0 are positive integers, and let τ denote the first time Zn hits a coordinate axis. We show E(τ) is finite if and only if α < 0, and in that case E(τ) equals i0j0/(−α). The analogous result holds for continuous-time martingales. Part of the theory carries over to higher dimensions, and we develop a class of processes which illustrate the limitations of the theory in that context