The distribution of the maximum of a second order autoregressive process: the continuous case

TL;DR

This paper derives the distribution of the maximum of a second order autoregressive process, providing explicit formulas and eigenvalue-based representations, especially for absolutely continuous variables, highlighting large deviations behavior.

Contribution

It offers a novel explicit distribution function for the maximum of AR(2) processes, including eigenvalue-based solutions and large deviations insights.

Findings

Distribution function expressed via eigenvalues and eigenfunctions

Explicit formulas for absolutely continuous variables

Large deviations expansion for maximum estimates

Abstract

We give the distribution function of $M_n$, the maximum of a sequence of $n$ observations from an autoregressive process of order 2. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. When the correlations are positive, P(M_n \leq x) =a_{n,x}, where a_{n,x}= \sum_{j=1}^\infty \beta_{jx} \nu_{jx}^{n} = O (\nu_{1x}^{n}), where $\{\nu_{jx}\}$ are the eigenvalues of a non-symmetric Fredholm kernel, and $\nu_{1x}$ is the eigenvalue of maximum magnitude. The weights $\beta_{jx}$ depend on the $j$th left and right eigenfunctions of the kernel. These results are large deviations expansions for estimates, since the maximum need not be standardized to have a limit. In fact such a limit need not exist.