Characterization of the simple cubic multivariate exponential families

The Letac–Mora class of real cubic natural exponential families has been characterized by a property of 2-orthogonality of an associated sequence of polynomials (see [G. Letac, M. Mora, Natural real exponential families with cubic variance functions, Ann. Statist. 18 (1990) 1–37; A. Hassairi, M. Zarai, Characterization of the cubic exponential families by orthogonality of polynomials, Ann. Probab. 32 (2004) 2463–2476]). The present paper introduces a notion of transorthogonality for a sequence of polynomial on Rd to extend the characterization to the multivariate version of the Letac–Mora class of real natural exponential families.