A New Generalization of Chebyshev Inequality for Random Vectors

In this article, we derive a new generalization of Chebyshev inequality for random vectors. We demonstrate that the new generalization is much less conservative than the classical generalization. 1 Classical Generalization of Chebyshev inequality The Chebyshev inequality discloses the fundamental relationship between the mean and variance of a random variable. Extensive research works have been devoted to its generalizations for random vectors. For example, various generalizations can be found in Marshall and Olkin (1960), Godwin (1955), Mallows (1956) and the references therein. A natural generalization of Chebyshev inequality is as follows. For a random vector X ∈ R with cumulative distribution F (.), Pr {||X − E[X]| | ≥ ε} ≤ Var(X) ε2 ∀ε > 0 (1) where ||.|| denotes the Euclidean norm of a vector and Var(X) def = ∫ V ∈ R ||V − E[X]||dF (V ) This classical generalization can be found in a number of textbooks of probability theory and statistics (see, e.g., pp. 446-451 of Laha and Rohatgi (1979)). ∗The author is with Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803; Email: [email protected].