Cooperative effects in risk models with discrete time

Abstract: In this paper risk models with small initial capital, insurance percent and ruin probability are constructed. These models may be used in different modern applications among which an insurance of a franchisee is one the most important. The models are based on a principle of a mutual insurance that is a considered system is an aggregation of a large number of identical insurance systems. We assume that these identical systems may be as independent so weak dependent. In such risk models phase transition phenomena are detected also. Main method to obtain these results is an estimate of rate convergence in limit theorems from probability theory.