On Divergence of the Saddlepoint Vector Field and Asymptotic Normality

Abstract

In this paper we shall investigate the state of convergence in law with a vector field of saddlepoint. The time fluctuation of these vector fields are visualized by 3-dim graphs. Through the perspective of fluid convergence, we can give a new interpretation that asymptotic normality is a phenomenon that the tangent vector space becomes uniform. If we extend the saddlepoint/inverse saddlepoint to a general dimensional space, it is pointed out that they have a Riemannian manifold structure in ℝ^d. And the tangent vector space of these manifolds are dual of each other. Especially the inverse saddlepoint manifold is a natural one which has a covariance matrix as the basis vector matrix of its tangent vector space, and an identification exists between the points and the normal distributions. These manifolds are attractive in a nonparametric approach to study the shape of probability distributions.