A uniform asymptotic estimate for ruin probability of a discrete-time risk model with subexponential innovations

The recursive equation T_n=X_n+T_n-1Y_n was considered, in which X_n and Y_n are two independent random variables, and T_n-1 on the right-hand side is independent of (X_n,Y_n). Under the assumption that X_n follows a subexponential distribution with a nonzero lower Karamata index, and that (X_n,Y_n) fulfills a certain dependence structure, some asymptotic formulas were obtained for the tail probabilities of T_n in this equation.