Large deviations for solutions to stochastic recurrence equations under Kesten's condition (Q359689)

Let \((Y_i)\) be the stationary solution to the stochastic recurrence equation \(Y_n=A_nY_{n-1}+B_n\), \(n\in \mathbb{Z}\), where \((A_i, B_i)\), \(i\in \mathbb{Z}\), is an i.i.d. sequence and \(A_i>0\)~a.s. In this paper, the authors prove large deviation results for the partial sum sequence \(S_n=Y_1+\cdots+Y_n\) under Kesten's condition, which are analogs to those obtained by \textit{A. V. Nagaev} [Theor. Probab. Appl. 14, 193--208 (1969); translation from Teor. Veroyatn. Primen. 14, 203--216 (1969; Zbl 0196.21003)] and \textit{S. V. Nagaev} [Ann. Probab. 7, 745--789 (1979; Zbl 0418.60033)] in the case of partial sums of i.i.d. random variables. By applying the large deviation results, the asymptotic behaviors of the ruin probabilities in the model are also considered.