  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 $M_0^\ast=0$.
