Ruin Theory And Subexponential Distribution

In the present paper, we survey probabilistic methods used in the classical risk model and renewal risk model paying particular attention to estimating the asymptotic ruin probability on the case where the condition for existence of adjustment coefficient does not hold. We give a detailed introduction of the class of subexpo-nential distribution and prove concisely that for a random walk with negative drift, the distribution of the ultimate maximum can be expressed as a compound geometric distribution. This result is of theoretical importance in the context of applied probability, often mentioned in some authentic references but rarely provided a detailed proof. The main purpose of this paper is to provide some theoretical basis for the research of a. class of heavy-tailed risk models on methodology. Furthermore, discrete subexponentiality in the fully discrete risk model has been found to yield some new results related to the asymptotic formula for the ultimate ruin probability in the paper.