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

Abstract

We give the cumulative distribution function of M _n , the maximum of a sequence of n observations from an autoregressive process of order 1. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. When the correlation is positive,

$$P \left( M_n \leq x \right)\ =a_{n,x},$$

where

$$a_{n,x}= \sum_{j=1}^\infty \beta_{jx}\ \nu_{jx}^{n} = O \left( \nu_{1x}^{n}\right),$$

where {ν _jx } are the eigenvalues of a non-symmetric Fredholm kernel, and ν _1x is the eigenvalue of maximum magnitude. When the correlation is negative

$$P \left( M_n \leq x \right)\ =a_{n,x} +a_{n-1,x}.$$

The weights β _jx depend on the jth left and right eigenfunctions of the kernel. These are given formally by left and right eigenvectors of an infinite Toeplitz matrix whose eigenvalues are just {ν _jx }. These results are large deviations expansions for extremes, since the maximum need not be standardized to have a limit. In fact, such a limit need not exist. The use of the derived expansion for P(M _n ≤ x) is illustrated using both simulated and real data sets.