Moment Inequalities for the Maximum Cumulative Sum

Assume $E(X_i) \equiv 0$. For $\nu \geqq 2$, bounds on the $\nu$th moment of $\max_{1 \leqq k \leqq n}|\sum^{a + k}_{a + 1} X_i|$ are deduced from assumed bounds on the $\nu$th moment of $|\sum^{a + n}_{a + 1} X_i|$. The inequality due to Rademacher-Mensov for $\nu = 2$ and orthogonal $X_i$'s is generalized to $\nu \geqq 2$ and other types of dependent $\operatorname{rv's}.$ In the case $\nu > 2$, a second result is obtained which is considerably stronger than the first for asymptotic applications.

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