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

Abstract

For every two-dimensional random walk on the square lattice Z 2 having zero mean and finite variance we obtain fine asymptotic estimates of the probability that the walk hits the negative real line for the first time at a site (s, 0), when it is started at a site far from both (0, s) and the origin.

The Hitting Distribution of a Line Segment for Two-Dimensional Random Walks

“…which entails the same property for ν in place of µ ( [13]: Theorem 1.1). (For more details see Appendix (C).…”

“….}. The proofs of Theorems 1 and 2 rest on the results on H ± x (s) obtained in [13] (Theorem 1.1; see also [14] for (9)) as given in the following theorem (and also in (23), (24) and (25) later).…”

“…The proof of Theorem 1 primarily rests on the asymptotic estimates obtained in [13] of hitting distributions for half-real-lines. The first visit of I(n) occurs after the consecutive overshoots in the first k alternating entrances of the walk X into the half-lines (−∞, −n) and (n, ∞) for some k = 0, 1, 2, .…”

“…The case δ = 0 is similarly dealt with based on the corresponding result for H − x (s) (Theorem 1 of [13]). Notice that 1/ √ x − n * ≤ 1/ √ y + n * if and only if y ≤ x − 2n * and that according to (II)…”

“…For the symmetric simple random walk (i.e., P 0 [S 1 = x] = 1/4 for x ∈ {±1, ±i}) we can improve the error estimate in [13] for H ± x (s) and accordingly that in Theorem 1 for…”