On Probability and Moment Inequalities for Supermartingales and Martingales

The probability inequality for $\max_{k\leq n}S_k$, where $S_k=\sum_{j=1}^kX_j$, is proved under the assumption that the sequence $S_k$, ${k=1,\ldots, n}$ is a supermartingale. This inequality is stated in terms of probabilities ${\bf P}(X_j>y) $ and conditional variances of random variables $X_j$, $j=1,\ldots, n$. As a simple consequence the well-known moment inequality due to Burkholder is deduced. Numerical bounds are given for constants in Burkholder's inequality.