ON PROBABILITY AND MOMENT INEQUALITIES FOR SUPERMARTINGALES AND MARTINGALES∗

(Translated by the author) Abstract. The probability inequality for maxk≤n Sk, where Sk = ∑k j=1Xj, is proved under the assumption that the sequence Sk, k = 1,..., n is a supermartingale. This inequality is stated in terms of probabilities P(Xj> y) and conditional variances of random variables Xj, j = 1,..., n. As a simple consequence the well-known moment inequality due to Burkholder is deduced. Numerical bounds are given for constants in Burkholder’s inequality