  In this paper we focus on the large n probability distribution function of
the largest eigenvalue in the Gaussian Orthogonal Ensemble of n by n matrices
(GOEn). We prove an Edgeworth type Theorem for the largest eigenvalue
probability distribution function of GOEn. The correction terms to the limiting
probability distribution are expressed in terms of the same Painleve II
functions appearing in the Tracy-Widom distribution. We conclude with a brief
discussion of the GSEn case.
