α-TRANSIENCE AND α-RECURRENCE FOR RANDOM WALKS AND LÉVY PROCESSES

Abstract

The authors investigate the α-transience and α-recurrence for random walks and Levy processes by means of the associated moment generating function, give a dichotomy theorem for not one-sided processes and prove that the process X is quasi-symmetric if and only if X is not α-recurrent for all α<0 which gives a probabilistic explanation of quasi-symmetry, a concept originated from C. J. Stone.