A characterization of multivariate distributions by conditional moments

摘要

Laha and Lukacs (1960) characterized distributions such that there exists a quadratic combination of i.i.d. random variables having a polynomial regression of order two on the sample mean. They obtain characterizations of Gaussian, Poisson, binomial, negative binomial, gamma and hyperbolic secant distributions. This article concerns an extension in terms of multidimensional i.i.d. random variables with distributions in natural exponential families. We first consider two i.i.d. random variables taking values in R~d, say U and V, and we write their sum as S = U + V. It is shown that there exists a ε R, and an affine transformation, L, such that E(U directX U|S) =a S directX S + L(S), if and only if the distribution of U and V is oneof those described in Casalis (1996): Poisson-Gaussian, multinomial, negative-multinomial-gamma, or hyperbolic distributions. In the case of more than two i.i.d. random variables, such a characterization is obtained by the conditional moments of some quadratic combination of the i.i.d. sample. The case of multidimensional process with stationary and independent increments is also considered. We also relate our results to a characterization due to Fosam and Shanbhag (1997). We then obtain a relation between multivariate distributions described in Hassairi (1990) and conditional moments.