Saddlepoint Approximations For Self-normalized Random Sums Without Moment Conditions

Self-normalized sum of random variables is closely related to t-statistics and other statistics.Therefore,it is an important issue in the theory of probability and statis-tics.The method of saddlepoint approximation can be used to approximate density,mass,distribution functions and related probability.Therefore,in this paper,we s-tudy the tail probability for the self-normalized random sums for strongly non-lattice random variables by saddlepoint approximations with no moment conditions.The counting processes employed include Poisson and renewal processes with Erlang(2)waiting time.An.asymptotic expansion for tail probability based on saddlepoint ap-proximation technique is derived for each case and tested in numerical experiments.and the true probability.The distributions for the tests include mixture of lattice and continuous,mixture of lattice and continuous with heavy tail,and continuous random variables.The result shows that our saddlepoint approximation formulas are uniformly more accurate than the naive normal approximations and the relative error between saddlepoint approximation and true value is small.