Bounds on the Moments of Martingales

Abstract

We prove the following THEOREM. Let $\{S_n, n \geqq 1\}$ be a martingale, $S_0 = 0, X_n = S_n - S_{n-1}, \gamma_{\nu n} = E(|X_n|^\nu)$ and $\beta_{\nu n} = (1/n) \sum^n_{j = 1} \gamma_{\nu j}$. Then for all $\nu\geqq 2$ and $n = 1, 2, \cdots$ \begin{equation*}\tag{1.1}E(|S_n|^\nu) \leqq C_\nu n^{\nu/2}\beta_{\nu n},\end{equation*} where \begin{equation*}\tag{1.2}C_\nu = \lbrack 8(\nu - 1) \max (1, 2^{\nu - 3})\rbrack^\nu.\end{equation*} As shown by Chung ([3], pp. 348-349) an inequality of Marcinkiewicz and Zygmund ([5], p. 87) implies that the theorem holds (possibly with a different value of $C_\nu$) whenever the $X$'s are independent. In the same way the above theorem is implied by the generalization of the Marcinkiewicz-Zygmund result given by Burkholder ([2], Theorem 9). However, our proof is elementary. Although our choice of $C_\nu$ is not the best possible, it is explicit. For the case of independent $X$'s, von Bahr ([6], p. 817) has given a bound for $E(|S_n|^\nu)$ which may sometimes involve powers of $\beta_{\nu n}$ higher than 1. Finally Doob ([4], Chapter V, Section 7) has treated the case when the $X$'s form a Markov chain. After proving some lemmata in Section 2, we give the proof of the theorem in Section 3. The case of exchangeable random variables is dealt with in Section 4.