Multivariate Chebyshev Inequalities

If X is a random variable with EX² = σ², then by Chebyshev's inequality, {equation*}{1.1}P\{|X| ε\} σ^2/ε^2.{equation*} If in addition $EX = 0$, one obtains a corresponding one-sided inequality {equation*}{1.2} P\{X ε\} σ^2/ (ε^2 + σ^2){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 ε = 1. There are many possible multivariate extensions of (1.1) and (1.2). Those providing bounds for Pmax1 j k |Xⱼ| 1\ and P\|max1 j k Xⱼ 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, ⋯, 1) by e and (0, ⋯, 0) by 0; the dimensionality will be clear from the context. If x = (x₁, ⋯, xₖ) and y = (y₁, ⋯, yₖ), we write x y(x > y) to mean xⱼ yⱼ(xⱼ > yⱼ), j = 1, 2, ⋯, k. If Σ = (σᵢⱼ): k × k is a moment matrix, for convenience we write σⱼⱼ = σ²ⱼ, j = 1, ⋯, k. Unless otherwise stated, we assume that Σ is positive definite.