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

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.