On a Szego Type Limit Theorem, the Holder-Young-Brascamp-Lieb Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of Stationary Fields

TL;DR

This paper explores advanced limit theorems for sums and quadratic forms of stationary processes, extending spectral and cumulant methods to multidimensional cases with applications in statistical analysis.

Contribution

It extends the functional analysis approach for central limit theorems to multidimensional indices and applies it to quadratic forms and spectral integrability in stationary fields.

Findings

Extended CLT framework to multidimensional indices.

Demonstrated applications to quadratic forms.

Clarified role of Appell polynomial expansion rank.

Abstract

Many statistical applications require establishing central limit theorems for sums, integrals, or for quadratic forms of functions of a stationary process. A particularly important case is that of Appell polynomials, since the Appell expansion rank" determines typically the type of central limit theorem satisfied by these functionals. We review and extend here to multidimensional indices a functional analysis approach to this problem proposed by Avram and Brown (1989), based on the method of cumulants and on integrability assumptions in the spectral domain; several applications are presented as well.