On the strong law of large numbers for sequences stationary in the narrow sense

Abstract

In his recent paper published in Vestnik St. Petersburg University, Ser. Mathematics, V.V. Petrov found new sufficient conditions for the fulfillment of the strong law of large numbers for sequences of random variables stationary in the broad sense. These conditions are expressed in terms of second moments. In this paper, by using the ergodic theorem, similar problems are solved for sequences of random variables stationary in the narrow sense. In the absence of second moments, the statements of conditions involve the truncated second moments of truncated random variables. At the end of the paper, an example of a stationary sequence of random variables which is not ergodic but obeys the strong law of large numbers is given.