Markov Processes

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For a general class of random matrices of size $n$ having independent entries, the typical eigenvalue is of order $\sqrt n$. We had investigated the large deviation properties when we are looking for eigenvalues that are of order $n$. We had looked at the case when the tails decay faster than Gaussian and now plan to examine the Gaussian case and we expect the results to be somewhat different.Large deviation theory deals with the estimation of probabilities of rare events. They are necessarily tiny and are often exponentially small in some natural parameter. The theory deals with estimating with some precision the exponential rate. When we study sums of random variables this often involves the estimation of the expectation of exponential moments of the sums. We have proposed to examine this phenomena in certain sums of unusual but general form that have applications to other contexts.