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

Abstract

We give the cumulative distribution function of \(M_n\), the maximum of a sequence of n observations from a first order moving average. Solutions are first given in terms of repeated integrals and then for the case, where the underlying independent random variables have an absolutely continuous probability density function. When the correlation is positive, $P( M_n \leq x ) \ =\ \sum\limits _{j=1}^{\infty } \beta _{j, x} \ \nu _{j, x}^{n},$where \(\{\nu _{j, x}\}\) are the eigenvalues (singular values) of a Fredholm kernel and \(\beta _{j, x}\) are some coefficients determined later. 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. 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 $\sum\limits _{i=1}^{\infty } w_i ( \lambda -\theta _i )^{-1}=\lambda -\theta _0$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.