The hitting distributions of a line for two dimensional random walks

Abstract

Abstract. For every irreducible random walk on Z 2 with zero mean and finite 2 + δ absolute moment (0 ≤ δ < 1) we obtain fine asymptotic estimates of the probability that the first visit of the walk to the horizontal axis takes place at a specified site of it.

The hitting distributions of a half real line for two-dimensional random walks

“…The present author [12] derived certain asymptotic expressions of H x+iy (s) as |x−s+iy|→∞ which are valid uniformly either in x−s or in y. In this paper we shall obtain similar ones for H − x+iy .…”

“…From the result on H − x given above together with those on H x+iy obtained in [12] one can derive asymptotic formulae for H − x+iy as we shall exhibit at the end of this section in the case when |x−s+iy|∧|x+iy|→∞.…”

“…We conclude this section by exhibiting estimates of H − x+iy (s) (for x+iy / ∈L + ) which are deduced from theorems given above combined with the estimates of H iy (s) in [12] in view of the decomposition…”

“…as well as the estimates of v n and u n and of H 0 in Lemma 3.3 and Proposition 3.4 and in [12], respectively.…”

“…where o(1) approaches zero as |x−s+iy|→∞ (Theorem 2 of [12]). Using this together with (3) (with x≥0) the Brownian version of (11) (see (50)) yields not only the formula (1) but also that for s<0, as |x−s+iy|∧|x+iy|→∞,…”