The Annals of Mathematical Statistics

Multivariate Chebyshev Inequalities

Albert W. Marshall and Ingram Olkin

Full-text: Open access

Abstract

If $X$ is a random variable with $EX^2 = \sigma^2$, then by Chebyshev's inequality, \begin{equation*}\tag{1.1}P\{|X| \geqq \epsilon\} \leqq \sigma^2/\epsilon^2.\end{equation*} If in addition $EX = 0$, one obtains a corresponding one-sided inequality \begin{equation*}\tag{1.2}\quad P\{X \geqq \epsilon\} \leqq \sigma^2/ (\epsilon^2 + \sigma^2)\end{equation*} (see, e.g., [8] p. 198). In each case a distribution for $X$ is known that results in equality, so that the bounds are sharp. By a change of variable we can take $\epsilon = 1$. There are many possible multivariate extensions of (1.1) and (1.2). Those providing bounds for $P\{\max_{1 \leqq j \leqq k} |X_j| \geqq 1\}$ and $P\{|\max_{1 \leqq j \leqq k} X_j \geqq 1\}$ have been investigated in [3, 5, 9] and [4], respectively. We consider here various inequalities involving (i) the minimum component or (ii) the product of the components of a random vector. Derivations and proofs of sharpness for these two classes of extensions show remarkable similarities. Some of each type occur as special cases of a general theorem in Section 3. Bounds are given under various assumptions concerning variances, covariances and independence. Notation. We denote the vector $(1, \cdots, 1)$ by $e$ and $(0, \cdots, 0)$ by 0; the dimensionality will be clear from the context. If $x = (x_1, \cdots, x_k)$ and $y = (y_1, \cdots, y_k)$, we write $x \geqq y(x > y)$ to mean $x_j \geqq y_j(x_j > y_j), j = 1, 2, \cdots, k$. If $\Sigma = (\sigma_{ij}): k \times k$ is a moment matrix, for convenience we write $\sigma_{jj} = \sigma^2_j, j = 1, \cdots, k$. Unless otherwise stated, we assume that $\Sigma$ is positive definite.

Article information

Source
Ann. Math. Statist., Volume 31, Number 4 (1960), 1001-1014.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177705673

Digital Object Identifier
doi:10.1214/aoms/1177705673

Mathematical Reviews number (MathSciNet)
MR119234

Zentralblatt MATH identifier
0244.60013

JSTOR
links.jstor.org

Citation

Marshall, Albert W.; Olkin, Ingram. Multivariate Chebyshev Inequalities. Ann. Math. Statist. 31 (1960), no. 4, 1001--1014. doi:10.1214/aoms/1177705673. https://projecteuclid.org/euclid.aoms/1177705673


Export citation