The distribution of the maximum of a first order moving average: the continuous case

Abstract

We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables have an absolutely continuous density. When the correlation is positive, $$ P(M_n %\max^n_{i=1} X_i \leq x) =\ \sum_{j=1}^\infty \beta_{jx} \nu_{jx}^{n} \approx B_{x} \nu_{1x}^{n} $$ where %$\{X_i\}$ is a moving average of order 1 with positive correlation, and $\{\nu_{jx}\}$ are the eigenvalues (singular values) of a Fredholm kernel and $\nu_{1x}$ is the eigenvalue of maximum magnitude. A similar result is given when the correlation is negative. The result is analogous to large deviations expansions for estimates, since the maximum need not be standardized to have a limit. % there are more terms, and $$P(M_n <x) \approx B'_{x}\ (1+\nu_{1x})^n.$$ For the continuous case the integral equations for the left and right eigenfunctions are converted to first order linear differential equations. The eigenvalues satisfy an equation of the form $$\sum_{i=1}^\infty w_i(\lambda-\theta_i)^{-1}=\lambda-\theta_0$$ for certain known weights $\{w_i\}$ and eigenvalues $\{\theta_i\}$ of a given matrix. This can be solved by truncating the sum to an increasing number of terms.