On the notion of recurrence in discrete stochastic processes

Received by the editors March 3, 1947. 1 John Simon Guggenheim Memorial Fellow. 2 That the theory of stationary stochastic processes is mathematically equivalent with an "ergodic" theory (to which one is also led by dynamical considerations) was clearly recognized by Doob in 1934. See J. L. Doob, Stochastic processes and statistics, Proc. Nat. Acad. Sci. U.S.A. vol. 20 (1934) pp. 376-379. The analogies discussed in the present paper are but particular cases of Doob's general equivalence principle. However, since the motivations underlying the statistical and the dynamical points of view are of a somewhat different physical character it seemed desirable to treat both cases separately.