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

TL;DR

This paper derives the distribution of the maximum of a first order moving average process, providing solutions in integral form and explicit formulas for discrete cases, highlighting eigenvalue-based approximations for positive correlation.

Contribution

It offers a novel explicit distribution formula for the maximum of a moving average process, especially in discrete cases, using eigenvalues of a related matrix.

Findings

Distribution expressed via eigenvalues of a matrix.

Approximate probability using dominant eigenvalue for large n.

Eigenvalues depend only on the range of the underlying variables.

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 are discrete. When the correlation is positive, $$ P(M_n \max^n_{i=1} X_i \leq x) = \sum_{j=1}^\infty \beta_{jx} \nu_{jx}^{n} \approx B_{x} r{1x}^{n} $$ where $\{\nu_{jx}\}$ are the eigenvalues of a certain matrix, $r_{1x}$ is the maximum magnitude of the eigenvalues, and $I$ depends on the number of possible values of the underlying random variables. The eigenvalues do not depend on $x$ only on its range.