H. Kesten et Ra. Maller, Stability and other limit laws for exit times of random walks from a stripor a halfplane, ANN IHP-PR, 35(6), 1999, pp. 685-734
Citations number
23
Categorie Soggetti
Mathematics
Journal title
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES
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.