On the Application of the Individual Ergodic Theorem to Discrete Stochastic Processes

That Birkhoff's individual ergodic theorem provides a strong law of large numbers for stationary stochastic processes has been known since its applications to this end by Doob (cf. [3; 4]) and Hopf [9]. The conclusion of the ergodic theorem as it applies to a sequence of random variables, {x " }, is from one point of view considerably stronger than that of the strong law of large numbers. The latter states (in part) that the sequence of arithmetic means {(l/£) Sf=ix»} converges with probability 1, while the individual ergodic theorem will be used in § §2 and 3 in giving necessary and sufficient conditions for the convergence with probability 1 of the sequence {(!/£) /kInzA where \zs\ is any of a wide class of sequences of random variables of the form z, =<p(xx+ " x2+ " ■ ■ ■) and where <p(xx, x2, • • •) is a measurable function on the infinite product space associated with the sequence {x " } (in particular, if 0(xi, x2, • • •)=x, then z, = xs+1). Studies by Dunford and Miller [7] and by Dowker [6] involving averaged measures suggested the approach used.