Pentadiagonal Matrices and an Application to the Centered MA(1) Stationary Gaussian Process

TL;DR

This paper analyzes the properties of a specific pentadiagonal matrix with perturbed corners, deriving explicit formulas for its positivity and determinant, and applies these results to the cumulant generating function of a centered MA(1) Gaussian process.

Contribution

It provides explicit formulas for the positivity and determinant of perturbed pentadiagonal matrices and applies these to the analysis of a centered MA(1) Gaussian process.

Findings

Explicit conditions for non-negativity and positive definiteness of the matrix.

Closed-form expression for the determinant of the matrix.

Explicit limiting cumulant generating function for the MA(1) process.

Abstract

In this work, we study the properties of a pentadiagonal symmetric matrix with perturbed corners. More specifically, we present explicit expressions for characterizing when this matrix is non-negative and positive definite in two special and important cases. We also give a closed expression for the determinant of such matrices. Previous works present the determinant in a recurrence form but not in an explicit one. As an application of these results, we also study the limiting cumulant generating function associated to the bivariate sequence of random vectors (n^{-1} (\sum_{k=1}^n X_k^2 , \sum_{k=2}^n X_k X_{k-1})_{n in N}, when (X_n)_{n in N} is the centered stationary moving average process of first order with Gaussian innovations. We exhibit the explicit expression of this limiting cumulant generating function. Finally, we present three examples illustrating the techniques studied here.