On the rate of growth of the overshoot and the maximum partial sum

Let T_r be the first time at which a random walk Sn escapes from the strip [−r, r], and let |ST_r| − r be the overshoot of the boundary of the strip. We investigate the order of magnitude of the overshoot, as r → ∞, by providing necessary and sufficient conditions for the ‘stability’ of |ST_r|, by which we mean that |ST_r|/r converges to 1, either in probability (weakly) or almost surely (strongly), as r → ∞. These also turn out to be equivalent to requiring only the boundedness of |ST_r|/r, rather than its convergence to 1, either in the weak or strong sense, as r → ∞. The almost sure characterisation turns out to be extremely simple to state and to apply: we have |ST_r|/r → 1 a.s. if and only if EX² < ∞ and EX = 0 or 0 < |EX| ≤ E|X| < ∞. Proving this requires establishing the equivalence of the stability of ST_r with certain dominance properties of the maximum partial sum S_n* = max(|S_j| : 1 ≤ j ≤ n) over its maximal increment.