Some Sharp Multivariate Tchebycheff Inequalities

Abstract

If $X$ is a random variable with $EX = 0, \operatorname{var}(X) = \sigma^2$, then the inequalities \begin{equation*}\begin{align*}\tag{1.1} P\lbrack | X | \geqq \epsilon\rbrack \leqq \sigma^2/\epsilon^2, \\ \tag{1.2}P\lbrack X \geqq \epsilon\rbrack \leqq \sigma^2/(\epsilon^2 + \sigma^2)\end{align*},\end{equation*} where $\epsilon > 0$, are known as, respectively, the Tchebycheff and the one-sided Tchebycheff inequalities. It may be noted that under the stated conditions (1.2) can always be attained and (1.1) can be attained if $\sigma^2 \leqq \epsilon^2$. Let $\mathbf{X} = (X_1, X_2, \cdots, X_n)$ be a random vector with $E\mathbf{X} = \mathbf{0}, E\mathbf{X'X} = \mathbf{\Sigma, T}_+$ be a closed convex region in $R^n$, and $T = T_+ \bigcup \{\mathbf{X} \mid - \mathbf{X} \epsilon T_+\}$. In [3], Marshall and Olkin have obtained sharp upper bounds on $P\lbrack\mathbf{X} \epsilon T\rbrack$ and $P\lbrack\mathbf{X} \epsilon T_+\rbrack$ as multivariate generalizations of (1.1) and (1.2). They have used these bounds to obtain explicit sharp bounds on $P\lbrack \min X_i \geqq 1\rbrack, P\lbrack \min | X_i| \geqq 1\rbrack, P\lbrack |\prod X_i| \geqq 1\rbrack, P\lbrack \prod X_i \geqq 1, X_i > 0, i = 1, 2, \cdots, n\rbrack$ etc., when only the variances $\sigma_i^2$ of $X_i, i = 1, 2, \cdots, n$, are known. Let $\mathbf{Y} = (Y_1, Y_2, \cdots, Y_n)$ be a vector of $n$ nonnegative random variables with $E\mathbf{Y} = u$ and $\varphi \geqq 0$ be a homogeneous concave function on the nonnegative orthant $R_+^n$ of $R^n$. In Section 2 we have proved the main inequality of this paper, $P\lbrack\varphi(\mathbf{Y}) \geqq \epsilon\rbrack \leqq \varphi(\mathbf{u})/ \epsilon, \quad \epsilon > 0$, which is attainably sharp if $\varphi(\mathbf{u}) \leqq \epsilon$. In Section 3 this result has been used to obtain various generalizations of (1.1) and (1.2) to $n$ jointly distributed random variables or random vectors, which are also generalizations of the inequalities due to Marshall and Olkin for the case where only variances are known. In Section 4 we have obtained some sharp probability inequalities for some functions of symmetric psd (positive semidefinite) random matrices and have discussed their relation to inequalities due to Mudholkar [4].