On Moments of the Maximum of Normed Partial Sums

Abstract

Let $X, X_1, X_2, \cdots$ be independent random variables with $E(X_n) = 0 (n \geqq 1)$, and put $S_n = X_1 + \cdots + X_n (n \geqq 1)$. Marcinkiewicz and Zygmund [5] and Wiener [8] have shown that if the $X$'s have a common distribution, then \begin{equation*}\tag{1}E\{\sup_n|S_n/n|\} < \infty\end{equation*} provided that \begin{equation*}\tag{2}E\{|X|U (|X|)\} < \infty,\end{equation*} where we have put $U(x) = \max (1, \log x) (U_2(x) = U(U(x))$, etc.). Burkholder [2] has extended this result by showing that (1), (2), and \begin{equation*}\tag{3}E\{\sup_n|X_n/n|\} < \infty,\end{equation*} are equivalent. More recently, motivated by certain optimal stopping problems Teicher [7] and Bickel [1] under various assumptions on the distributions of $X_1, X_2, \cdots$ have shown that \begin{equation*}\tag{4}E\{\sup_n c_n|S_n|^\alpha\} < \infty\end{equation*} for certain sequences $(c_n)$ and positive constants $\alpha$. The interesting special case \begin{equation*}\tag{5}c_n = (nU_2(n))^{-\alpha/2}\end{equation*} is not covered by the results of these authors. This note gives a method which seems suitable for proving statements like (4) in a variety of cases. The method involves modifications of standard techniques used in the study of the law of the iterated logarithm. In particular, for each $\alpha = 1, 2, \cdots$ we are able to establish necessary and sufficient conditions for (4) when the $X$'s are identically distributed and the sequence $(c_n)$ satisfies (5). In Section 2 we state and prove one such theorem. Section 3 is devoted to explaining in somewhat more detail the scope of our results and their relation to the previously mentioned literature.