**Sankhya:
The Indian Journal of Statistics**

2004, Volume 66, Pt. 1, 1--19

**A Characterization of Lancaster
Probabilities with Margins in a Multivariate Additive Class**

By

D. Pommeret, Crest-Ensai, France

SUMMARY. Let $X$ and $Y$ be two random variables on $\R^d$ and let $(P_n)_{n\in\N^d}$ and $(Q_k)_{k\in\N^d}$ be two basis of orthonormal polynomials with respect to the distributions of $X$ and $Y$, respectively. The joint distribution of $(X,Y)$ is called a {\it Lancaster probability} if the expectation $\E(P_n(X)Q_k(Y))$ vanishes for $ n \neq k$. This paper concerns the characterization of Lancaster probabilities for the particular case $X=U+V$ and $Y=V+W$, where $U,V,W$ are independent random variables on $\R^d$. It is shown that the distribution of $(X, Y)$ is a Lancaster probability if and only if the natural exponential families generated by the random variables $U, V, W$ are simple quadratic (the variance is a specific quadratic function of the mean). This result is an extension of Lancaster (1975). We also generalize the definition of Lancaster probabilities that we relate to the more general class of quadratic natural exponential families on $\R^d$. Finally, two tests for independence and a goodness of fit test for these multivariate joint distributions are outlined.

*AMS (1991) subject classification}. *62H05, 62E10.

*Key words and phrases. *Lancaster probabilities, orthogonal
polynomials, quadratic and simple quadratic natural exponential families.