On the Strong Law of Large Numbers and Maximal Inequalities for Sums of Random Variables

We obtain new inequalities for the moments of maximum moduli of sums of random variables without any assumptions on the type of dependence for the summands. In the paper, the conditions are expressed in terms of bounds for the absolute moments of increments of the sums of the random variables under consideration. These conditions are more general than those used by other authors. There are examples of random variables that satisfy these conditions and do not satisfy the conditions of earlier results. Using the maximal inequalities, we prove the strong law of large numbers and derive a Baum–Katz type bound for the convergence rate in this law. Moreover, we prove a complete convergence result for normalized sums of random variables.