Precise Large Deviations For The Difference Of Two Sums Of Random Variables

With rapid development of the financial industry,insurance has become a hot spot of the current society.Reality prompted the academic knowledge of insurance industry to a higher level.In this area,due to the wide appliance in real life,the precise large deviation attracted a lot of renowned researchers.Back to1987.a famous mathematician called Bingham with some other researchers achieved some outstanding results which served as the cornerstone of the precise large deviation. During the last decade.some mathemati-cians made a few breakthroughs.Among these researchers,Wang and Tang are the most outstanding ones. In Wang(2007),he extended the classic result P(Sn—ESn＞x) nn(x),as n→∝tp P(S(k:n1,...,nk)—∑i=1k niμi＞x)～∑i=1k niFi(x)which could be considered as a pioneer work in the area of precise large deviation.Assume that there are two types of insurance contracts in an insurance company. The jth related claims are denoted by{xij,j≥1},i=1,2.In this paper we inves-tigate large deviations for the difference∑j=1n1(t) X1j—∑j=1n2(t) X2j and random difference∑j=1N1(t) X1j—∑j=1N2(t) X2j,where ni(t)are positive integer functions as t→∞,ni(t)∝,i=1,2.Ni(t),i=1,2are counting processes for the claim number.