Independence times for iid sequences, random walks and Levy processes

For a sequence in discrete time having stationary independent values (respectively, random walk) X, those random times R of X are characterized set-theoretically, for which the strict post-R sequence (respectively, the process of the increments of X after R) is independent of the history up to R. For a Levy process X and a random time R of X, reasonably useful sufficient conditions and a partial necessary condition on R are given, for the process of the increments of X after R to be independent of the history up to R. (C) 2018 Elsevier B.V. All rights reserved.