Some estimates for sums of dependent random variables which hold almost surely

Some estimates are proved for sums of dependent random variables. Theorem 1 contains no assumptions regarding the existence of moments of the random variables. In Theorem 2 estimates are given for the growth of sums of random variables in a stationary sequence.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 113–116, 1976.