# A matrix version of Chernoff inequality

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## Abstract

An interesting result from the point of view of upper variance bounds is the inequality of Chernoff [Chernoff, H., 1981. A note on an inequality involving the normal distribution. Annals of Probability 9, 533-535]. Namely, that for every absolutely continuous function g with derivative g' such that , and for standard normal r.v. [xi], . Both the usefulness and simplicity of this inequality have generated a great deal of extensions, as well as alternative proofs. Particularly, Olkin and Shepp [Olkin, I., Shepp, L., 2005. A matrix variance inequality. Journal of Statistical Planning and Inference 130, 351-358] obtained an inequality for the covariance matrix of k functions. However, all the previous papers have focused on univariate function and univariate random variable. We provide here a covariance matrix inequality for multivariate function of multivariate normal distribution.

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