This paper discusses the convergence of a tail probability series on U-statistic.

For the partial sums S_n of pairwise NQD sequences,by establishing its tail probability inequality,its upper bounds of p-order moment and tail probability were obtained,and,under different distributions,a sufficient condition for WLLN was studied.

Based on extreme value theory and methods,generalized Pareto models are established to extreme daily return to Shanghai stock index,and return levels and upper tail probability are worked out by using the models.

When people invest the financial assets,the primary problem which investors pay attention to is often the large fluctuation probability of financial asset return,for short tail probability.

d(independent and identically distributed) random variables and Chernoff bound are reviewed,with which the tail probability of n i.

Under the assumption that X is subexponentially distributed, we obtained an asymptotics of the tail probabilities of the maxima of radom partial sums.

In chapter one, we discuss the limit behaviour of a kind of symmetric random variables sequences, mainly including the strong law of large numbers, the convergence rates of tail probabilities in the law of large numbers and the convergence rates of tail probabilities in the law of large numbers with random index.