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 S_n escapes from the strip [-r, r], and let |S_Tr| - 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 |S_Tr|, by which we mean that \S_Tr|/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 |S_Tr|/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 |S_Tr|/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 S_Tr with certain dominance properties of the maximum partial sum S_n^* = max{|S_j| : 1 ≤ j ≤ n} over its maximal increment.