Abstract
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.
Citation
R. J. Serfling. "Moment Inequalities for the Maximum Cumulative Sum." Ann. Math. Statist. 41 (4) 1227 - 1234, August, 1970. https://doi.org/10.1214/aoms/1177696898
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