Almost sure relative stability of the overshoot of power law boundaries

We give necessary and sufficient conditions for the almost sure relative stability of the overshoot of a random walk when it exits from a two-sided symmetric region with curved boundaries. The boundaries are of power-law type, ±rn ^b , r > 0, n = 1, 2,..., where 0 ≤ b < 1, b ≡ 1/2. In these cases, the a.s. stability occurs if and only if the mean step length of the random walk is finite and non-zero, or the step length has a finite variance and mean zero.