On Compactness Properties of the Exit Position of a Random Walk From an Interval

We study the exit position S T(r) of a random walk S n from the interval [− r , r ], showing that the tightness of | S T(r) |/ r is equivalent to a generalised kind of stochastic compactness of S n which we call SC″ . This property is in turn equivalent to another kind of compactness property, which we call SC″ , of the maximal sum S n * = max 1 ⩽ j ⩽ n | S j |. The classes SC′ and SC″ , and a related class SC 0 , which so far seem unexplored, are related to, but different from, the class of stochastically compact S n studied by Feller, and are similarly of interest in the study of the weak convergence properties of S n and S T(r) . We give equivalent characterisations of SC′ and SC″ in terms of the domination of S n and S n * over their maximal increment, and also some analytic characterisations in terms of functionals of the underlying distribution. As a corollary we obtain an equivalence for the stochastic compactness of | S T(r) |/ r . 1991 Mathematics Subject Classification : primary 60K05, 60J15, 60F05; secondary 60G40, 60G50.