An explicit representation of Verblunsky coefficients

TL;DR

This paper presents a new explicit representation of Verblunsky coefficients for stationary processes, simplifying previous formulas and enabling precise analysis of their asymptotic behavior, especially for short-memory and FARIMA processes.

Contribution

It introduces a non-fractional explicit formula for Verblunsky coefficients, improving upon previous representations and facilitating detailed asymptotic analysis.

Findings

Derived a simpler explicit formula for Verblunsky coefficients

Provided estimates for short-memory processes

Determined asymptotic behavior for FARIMA processes

Abstract

We prove a representation of the partial autocorrelation function (PACF) of a stationary process, or of the Verblunsky coefficients of its normalized spectral measure, in terms of the Fourier coefficients of the phase function. It is not of fractional form, whence simpler than the existing one obtained by the second author. We apply it to show a general estimate on the Verblunsky coefficients for short-memory processes as well as the precise asymptotic behaviour, with remainder term, of those for FARIMA processes.