The Maximum Term and First Passage Times for Autoregressions

Abstract

The limiting distribution of the maximum term of the non-normal stationary sequence $\cdots X_{-1}, X_0, X_1 \cdots$ satisfying the autoregressive equation $X_n = \varepsilon_n + a_1X_{n-1} + a_2X_{n-2} + \cdots$ is investigated when $\sum |a_k| < 1$ and $\cdots \varepsilon_{-1}, \varepsilon_0, \varepsilon_1 \cdots$ are integrable real valued i.i.d. random variables having distributions with tails that are either Pareto or exponential in nature. Asymptotic results for the joint distribution of the first passage time $t = \inf\{n: X_n \geq c\}$ and the excess $R_t = X_t - c$ are also given as $c \rightarrow \infty$.