Some Asymptotics Of Counting Processes In Insurance And Finance

It is just like what Embrechts, Kl ppelberg and Mikosch(1997)[6] pointed out: " random sums are the bread and butter of insurance mathematics ". And what have intimate relation with random sums are renewal counting processes and other counting processes. In the current theory of insurance and finance, the random variables generating counting processes are often supposed independent identically distributed, This hypothesis is reasonable in many situations and we have obtained rather satisfactory' results about it. But in more situations the random variables generating counting processes may not independent identically distributed, and in all kinds of dependent relations, negative association(NA) and positive association(PA) are commonly seen . The research and apply in this aspect are rather valuable. In Chap 2 we prove Wald inequalities and fundamental renewal theorems of renewal counting processes generated by NA sequences and PA sequences; In Chap 3 we are enlightened by Cheng and Wang[8], extend some results in Gut and Steinebach[7], obtain the precise asymptotics for renewal counting processes and depict the convergence rate and limit value of renewal counting processes precisely; At last,in the study of NA sequences, Su,Zhao and Wang(1996) [9], Lin(1997)[10] have proved the weak convergence for partial sums of stong stationary NA sequences.However product sums are the generalization of partial sums and also the special condition of more general U-statistic . The two kinds of sums have close relation and also have essential difference. So we discuss the weak convergence for product sums of stong stationary NA sequences in Chap 4. Because counting processes are also a kind of partial sums and can generate product sums. This result prepares for the study of weak convergence of counting processes.