Bounds on Moments of Certain Random Variables

Abstract

Let $\{X_n, n \geqq 1\}$ be a sequence of random variables and let $S_n = \sum^n_{i=1} X_i$. Under the condition that $\{S_n\}$ forms a martingale sequence, it was shown in [2] that, for $\nu \geqq 2$, $E(|S_n)^\nu) \leqq C_\nu n^{(\nu/2)-1} \sum^n_{i=1} E|X_i|^\nu,$ where \begin{equation*} \tag{(1)} C_\nu = \lbrack 8(\nu - 1) \max (1, 2^{\nu-3}) \rbrack^\nu.\end{equation*} The purpose of this paper is to show that the constant $C_\nu$ can be replaced by a much smaller constant in the following two cases: (i) $\nu$ is an even integer and the martingale dependence condition is replaced by one which is more explicit in terms of moments (Theorem 1); (ii) the $X_n$'s are independent with zero means (Theorem 2). For case (i) we give for $E(|S_n|^\nu)$ a bound which is a polynomial in $n$. This last bound does not appear to be too exhorbitant because, as shown by an example, it is not valid for all martingales $\{S_n\}$.*