Holonomic modules associated with multivariate normal probabilities of polyhedra

Abstract: The probability content of a convex polyhedron with a multivariate normal distribution can be regarded as a real analytic function. We give a system of linear partial differential equations with polynomial coefficients for the function and show that the system induces a holonomic module. The rank of the holonomic module is equal to the number of nonempty faces of the convex polyhedron, and we provide an explicit Pfaffian equation (an integrable connection) that is associated with the holonomic module. These are generalizations of results for the Schl\"afli function that were given by Aomoto.