Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GOE

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.