Given a joint probability density function of $N$ real random variables,$\{x_j\}_{j=1}^{N},$ obtained from the eigenvector-eigenvalue decomposition of$N\times N$ random matrices, one constructs a random variable, the linearstatistics, defined by the sum of smooth functions evaluated at the eigenvaluesor singular values of the random matrix, namely, $\sum_{j=1}^{N}F(x_j).$ Forthe jpdfs obtained from the Gaussian and Laguerre ensembles, we compute, inthis paper the moment generating function $\mathbb{E}_{\beta}({\rmexp}(-\lambda\sum_{j}F(x_j))),$ where $\mathbb{E}_{\beta}$ denotes expectationvalue over the Orthogonal ($\beta=1$) and Symplectic ($\beta=4)$ ensembles, inthe form one plus a Schwartz function, none vanishing over $\mathbb{R}$ for theGaussian ensembles and $\mathbb{R}^+$ for the Laguerre ensembles. These areultimately expressed in the form of the determinants of identity plus a scalaroperator, from which we obtained the large $N$ asymptotic of the linearstatistics from suitably scaled $F(\cdot).$
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