Finite generation of cumulants

In case of a finite state space, we can analyze statistical exponential families with tools from both Differential Geometry and Commutative Algebra, see e.g. Gibilisco, Riccomagno, Rogantin, Wynn eds (2009). A key feature of the algebraic framework is the finite generation of polinomial ideals. In the general case, the reduction to some sort of finite generation is more difficult, but it is a classical ingredient of standard statistical models. We suggested in Pistone and Wynn Statistica Sinica (1999) a definition of finite generation which is a generalization of the Morris class of distribution, see Morris Annals of Statistics (1982, 1983). We present here a development of this theory which makes use of recent results on Gröbner bases for D-modules. This is a joint work with Henry Wynn, LSE London UK.