Advancing gradually expansion of ruin probability in case of second-order regular variation tails

Ruin probability is an important scale of measurement for insurance company, for it can know own compensation ability well, so it is important to research ruin probability's advancing expansion for its stable manage. In this paper, we consider the classical risk model, assuming that the tail of claim-size is second-order regular variation. First we prove the closeness of second-order regular variation, we obtain advancing gradually expansion of equation, then using Beekman's convolution formula, finally get the advancing gradually expansion of ruin probability.