Hitting time of a half-line by two-dimensional random walk

Abstract

We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line {(x 1 , x 2 )|x 1 ≤ 0, x 2 = 0} before time n. It is proved that for aperiodic random walk with mean zero and finite 2 + δ(> 2)-th absolute moment, this probability times n 1/4 converges to some positive constant c * as n → ∞. We show that c * is expressed by using the characteristic function of the increment of the random walk. For the simple random walk, this expression gives c * = 1 + √ 2/(2 (3/4)).

Hitting Time of a Half-Line by a Two-Dimensional Non-Symmetric Random Walk

“…In [2], we deduce the asymptotic behavior of P 0 {T λ < τ V − } as λ ↑ 1 from that of P 0 {T λ < τ V − } × P 0 {T λ < τ V + } and P 0 {T λ < τ V − }/P 0 {T λ < τ V + }. The method to control P 0 {T λ < τ V − } follows from [2].…”

“…(2) When S is aperiodic with E[X] = 0 and E[|X| δ ] < ∞ for some δ > 0, it is proved that n 1/4 P 0 {τ V − > n} converges to some positive constant as n → ∞ (see [2]). Although Theorem 1.1 provides a weaker result for this model than that in [2], it has the advantage that it can be applied to a large class of random walks.…”

“…Although Theorem 1.1 provides a weaker result for this model than that in [2], it has the advantage that it can be applied to a large class of random walks.…”

“…In Section 2, we quote some results from [2]. In Section 3, we state some results needed to prove Theorem 1.1.…”

“…The proof of Lemma 2.3 is identical to that given in Section 17, Proposition 5 of [7] and is thus omitted (see [2,Proposition 2.4]). …”