Two-dimensional asymptotic expansions for large deviations of spherically distributed random vectors if the dominating point degenerates asymptotically

Abstract

Abstract. A geometric approach to asymptotic expansions for large-deviation probabilities, developed for the Gaussian law by Breitung and Richter [J. Multivariate Anal., 58, 1-20 (1996)], will be extended in the present paper to the class of spherical measures by utilizing their common geometric properties. This approach consists of rewriting the probabilities under consideration as large parameter values of the Laplace transfoma of a suitably defined function, expanding this function in a power series, and then applying Watson's lemma. A geometric representation of the Laplace transform allows one to combine the global and local properties of both the underlying measure and the large-deviation domain. A special new type of difficulty is to be dealt with because the so-called dominating points of the large-deviation domain degenerate asymptotically. As is shown in Richter and Schumacher (in print), the typical statistical applications of large-deviation theory lead to such situations. In the present paper, consideration is restricted to a certain two-dimensional domain of large-deviations having asymptotically degenerating dominating points. The key assumption is a parametrized expansion for the inverse ~-1 of the negative logarithm of the density-generating function of the two-dimensional spherical law under consideration,