Stability and other limit laws for exit times of random walks from a stripor a halfplane

We show that the passage time, T*(r), of a random walk S-n above a horizont al boundary at r (r greater than or equal to 0) is stable (in probability) in the sense that T*(r)/C(r) --> 1 as r --> infinity for a deterministic fu nction C(r) > 0, if and only if the random walk is relatively stable in the sense that S-n/B-n --> 1 as n --> infinity for a deterministic sequence B- n > 0. The stability of a passage time is an important ingredient in some p roofs in sequential analysis, where it arises during applications of Anscom be's Theorem. We also prove a counterpart for the almost sure stability of T*(r), which we show is equivalent to E/X/ < infinity, EX > 0. Similarly, c ounterparts for the exit of the random walk from the strip {/y/ less than o r equal to r} are proved. The conditions are further related to the relativ e stability of the maximal sum and the maximum modulus of the sums. Another result shows that the exit position of the random walk outside the boundar ies at +/-r drifts to infinity as r --> infinity if and only if the random walk drifts to infinity. (C) Elsevier, Paris.