Large deviation estimates for exceedance times of perpetuity sequences and their dual processes

In a variety of problems in pure and applied probability, it is of relevant to study the large exceedance probabilities of the perpetuity sequence \(Y_n := B_1 + A_1 B_2 + \cdots + (A_1 \cdots A_{n-1}) B_n\), where \((A_i,B_i) \subset (0,\infty) \times {\mathbb R}\). Estimates for the stationary tail distribution of \(\{ Y_n \}\) have been developed in the seminal papers of Kesten (1973) and Goldie (1991). Specifically, it is well-known that if \(M := \sup_n Y_n\), then \({\mathbb P} \left\{ M > u \right\} \sim {\cal C}_M u^{-\xi}\) as \(u \to \infty\). While much attention has been focused on extending this estimate, and related estimates, to more general processes, little work has been devoted to understanding the path behavior of these processes. In this paper, we derive sharp asymptotic estimates for the large exceedance times of \(\{ Y_n \}\). Letting \(T_u := (\log\, u)^{-1} \inf\{n: Y_n > u \}\) denote the normalized first passage time, we study \({\mathbb P} \left\{ T_u \in G \right\}\) as \(u \to \infty\) for sets \(G \subset [0,\infty)\). We show, first, that the scaled sequence \(\{ T_u \}\) converges in probability to a certain constant \(\rho > 0\). Moreover, if \(G \cap [0,\rho] \not= \emptyset\), then \({\mathbb P} \left\{ T_u \in G \right\} u^{I(G)} \to C(G)\) as \(u \to \infty\) for some "rate function" \(I\) and constant \(C(G)\). On the other hand, if \(G \cap [0,\rho] = \emptyset\), then we show that the tail behavior is actually quite complex, and different asymptotic regimes are possible. We conclude by extending our results to the corresponding forward process, understood in the sense of Letac (1986), namely, the reflected process \(M_n^\ast := \max\{ A_n M_{n-1}^\ast + B_n, 0 \}\) for \(n \in {\mathbb N}\), where \(M_0^\ast=0\).