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.
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Doney, R.A., Maller, R.A. Almost Sure Relative Stability of the Overshoot of Power Law Boundaries. J Theor Probab 20, 47–63 (2007). https://doi.org/10.1007/s10959-006-0040-3
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DOI: https://doi.org/10.1007/s10959-006-0040-3