© 1999 by London Mathematical Society
On Compactness Properties of the Exit Position of a Random Walk From an Interval
Department of Mathematics, Syracuse University Syracuse, NY 13244–1150, U.S.A. E-mail: pgriffin{at}math.syr.edu
Department of Mathematics, The University of Western Australia Nedlands 6907, Western Australia
Received 20 May 1997. Revision received 17 February 1998.
We study the exit position ST(r) of a random walk Sn from the interval [–r, r], showing that the tightness of |ST(r)|/r is equivalent to a generalised kind of stochastic compactness of Sn which we call SC''.
This property is in turn equivalent to another kind of compactness property, which we call SC'', of the maximal sum Sn* = max1 
j n |Sj|.
The classes SC' and SC'', and a related class SC0, which so far seem unexplored, are related to, but different from, the class of stochastically compact Sn studied by Feller, and are similarly of interest in the study of the weak convergence properties of Sn and ST(r).
We give equivalent characterisations of SC' and SC'' in terms of the domination of Sn and Sn* 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 |ST(r)|/r. 1991 Mathematics Subject Classification: primary 60K05, 60J15, 60F05; secondary 60G40, 60G50.
Present address: Department of Mathematics, University of Manchester, Manchester, M13 9 PL. E-mail: maller{at}ma.man.ac.uk