A Best Possible Kolmogoroff-Type Inequality for Martingales and a Characteristic Property

Abstract

In this paper some Kolmogoroff-type inequalities are derived. One of them is best possible for martingales. Another holds for sums of independent random variables and for martingales in which conditional variances satisfy a certain property. But it does not hold, in general, for martingales. This elucidates an important difference between martingales and independent sums. Let $E$ denote expectation. Consider a sequence of random variables $X_1, \cdots, X_n$ and the following three assumptions: \begin{equation*}\tag{a}E(X_1) = 0, E(X_i\mid X_{i-1}, \cdots, X_1) = 0,\quad i = 2, \cdots, n.\end{equation*}\begin{equation*}\tag{b}|X_i| \leqq T \text{almost surely},\quad i = 1, \cdots, n.\end{equation*}\begin{equation*}\tag{c} E(X_1^2) \neq 0, E(X_i^2\mid X_{i-1}, \cdots, X_1) \neq 0 \text{almost surely}, i = 2, \cdots, n.\end{equation*} The first says that $\{X_i\}$ is absolutely fair, the second that $\{X_i\}$ is almost surely bounded, and the third, that conditional variances are positive. We shall deal with the following classes of sums. $M(n)$ is the class of all martingales $\{S_i\}$ of $n$ partial sums $S_i = X_1 + \cdots + X_i$ where $\{X_i\}$ satisfies (a). $B(n) \subset M(n)$ is the subclass in which $\{X_i\}$ satisfies (a), (b), (c), $V(n) \subset B(n)$ the subclass where, additionally, $E(S_n^2) = E(S_n^2\mid X_{n-1}, \cdots, X_1)$, and finally $I(n) \subset B(n)$ the subclass where, in addition, $\{X_i\}$ are independent. $M(n)$ is a pneumonic for martingale, $B(n)$ for bounded martingale, $V(n)$ for martingales where variance equals conditional variance, and $I(n)$ for independent sums. Write $\sigma_i^2 = E(X_i^2), s_i^2 = \sigma_1^2 + \cdots + \sigma_i^2, i = 1, \cdots, n, C_1^2 = E(X_1^2), C_i^2 = E(X_i^2\mid X_{i-1}, \cdots, X_1), i = 2, \cdots, n, C^2 = C_1^2 + \cdots + C_n^2$. Let $M_n = \max (S_1, \cdots, S_n), p(t) = \mathrm{Pr}\{M_n \geqq ts_n^2\}$ and $r(t) = \lbrack e/(1 + tT)\rbrack^{ts_n^2/T}\lbrack 1/(1 + tT)\rbrack^{s_n^2/T^2}.$ Bennett [1] and Hoeffding [4] showed independently that $\sup_{I(n)} \lbrack\mathrm{Pr}\{S_n \geqq ts_n^2\}\rbrack \leqq r(t).$ Steiger [7] extended this to the context of Kolmogoroff inequalities by showing that $p(t) \leqq r(t)$ for all sums $\{S_i\} \varepsilon I(n)$. In Section 3 we show that the inequality actually holds throughout the larger class, $V(n)$, and that it is best-possible there. However the inequality can be false in $B(n)$. This shows that the maxima of martingale partial sums may have larger tail probabilities than those of independent sums or of sums in $V(n) - a$ characteristic property. In Section 2 we prove two preliminary lemmas of independent interest. These are used in Section 3 to obtain Theorem 1, which gives a best-possible upper bound in $B(n)$ for $\operatorname{Pr} \{M_n \geqq tC^2\}$ which, when restricted to $V(n)$ gives $\sup_{V(n)} \lbrack p(t)\rbrack\leqq r(t)$, also best possible. These results are compared to known results of Marshall [6], Dubins and Savage [2], and Steiger [7]. Finally, in Section 4, a partial converse to Theorem 1 is given.