Limit Theorems for Random Walk in a Half-Plane with Jump across the Border

The article is a continuation of the work [2], in which we obtained the analytical formulas for the probability of reaching the boundary by the random walk method on the integer points of the half-plane and for the probability of a jump across the border. In this paper we have found asymptotic approximations for the given probability distributions. These approximations are of special interest for using them in applications. Limit theorems for subcritical and supercritical cases lead to the normal law for the exit point or the jump across the border point, provided that the random walk stop occurred. In the critical case the asymptotic approach is different from the normal law. We obtained a stable distribution with α = 1/2. Limit theorems generalize the known special case, when there is no jump across the border. To derive the limit theorems, we applied the method of characteristic functions and Laplace transform method.

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