Laplace approximations for sums of independent random vectors. II: Degenerate maxima and manifolds of maxima
We consider expressions of the form \(Z_ n=E(\exp (nF(S_ n/n))\) where \(S_ n\) is the sum of n i.i.d. random vectors with values in a Banach space and F is a smooth real valued function. By results of Donsker- Varadhan and Bahadur-Zabell one knows that lim(1/n)log \(Z_ n=\sup _ x(F(x)-h(x))b_ F\) where h is the so-called entropy function. In part I [ibid. 72, 305-318 (1986; Zbl 0572.60007)] a more precise evaluation of \(Z_ n\) is given in the case where there was a unique point maximizing F-h and the curvature at the maximum was nonvanishing. The present paper treats the more delicate problem where these conditions fail to hold.
scientific article; zbMATH DE number 3984206
scientific article