A CHARACTERIZATION OF NORMAL DISTRIBUTION

Louis H. Y. Chen proved an inequality for the multivariate normal distribution N（0, I_n）in 1. In the present paper, we generlize this result, and characterize the multivariate normal distribution N （u, ∑） by the generalized inequality. Theorem: Suppose X=（X₁, …, X_n）' is a random vector with EX=u, Cov（X）=∑=(σ_ij≥0, then for any first-order partial diffentiable real function g on Rⁿ,\operatornameVarg(\boldsymbolX) \leqslant \sum_i=1^n \sum_j=1^n \sigma_i j E_x_i, g_z^\prime If and only if X～N （u, ∑）.