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In this paper, we study the large $N$ behavior of the moment-generatingfunction (MGF) of the linear statistics of $N\times N$ Hermitian matrices inthe Gaussian unitary, symplectic, orthogonal ensembles (GUE, GSE, GOE) andLaguerre unitary, symplectic, orthogonal ensembles (LUE, LSE, LOE) at the edgeof the spectrum. From the finite $N$ Fredholm determinant expression of the MGFof the linear statistics, we find the large $N$ asymptotics of the MGFassociated with the Airy kernel in these Gaussian and Laguerre ensembles. Thenwe obtain the mean and variance of the suitably scaled linear statistics. Weshow that there is an equivalence between the large $N$ behavior of the MGF ofthe scaled linear statistics in Gaussian and Laguerre ensembles, which leads tothe statistical equivalence between the mean and variance of suitably scaledlinear statistics in Gaussian and Laguerre ensembles. In the end, we use theCoulomb fluid method to obtain the mean and variance of another type of linearstatistics in GUE, which reproduces the result of Basor and Widom.

Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel