Markov Processes

对于大小为$n$的具有独立元素的一般随机矩阵，其典型特征值为$\sqrt n$阶。当我们寻找n阶的特征值时我们已经研究了大偏差的性质。我们已经研究了尾部衰减快于高斯分布的情况，现在我们计划研究高斯分布的情况，我们预计结果会有所不同。大偏差理论研究的是对罕见事件概率的估计。它们必然很小，在某些自然参数下往往是指数级小。该理论涉及对指数率进行一定精度的估计。当我们研究随机变量的和时，这通常涉及到对和的指数矩的期望的估计。我们建议以某些不寻常但普遍的形式来研究这一现象，这些形式适用于其他情况。

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.