On the Laplace Method for Gaussian Measures in a Banach Space

Abstract

In this paper, we prove results on sharp asymptotics for the probabilities $ P_A(uD) $, as $ u \to \infty $, where $ P_A $ is the Gaussian measure in an infinite-dimensional Banach space $ B $ with zero mean and nondegenerate covariance operator $ A $, $ D = \{x \in B: Q(x) \ge 0 \} $ is a Borel set in $ B $, and $Q$ is a smooth function. We analyze the case where the action functional attains its minimum on some set $ D $ on a one-dimensional manifold. We make use of the Laplace method in Banach spaces for Gaussian measures. Based on the general result obtained, for $ 0 < p \le 6 $ we find a sharp asymptotics for large deviations of distributions of $ L^p$-functionals for the centered Brownian bridge which arises as the limit while studying the Watson statistics. Explicit constants are given for the cases $ p = 1 $ and $ p = 2 $.