# Principal components analysis of regularly varying functions

## TL;DR

This paper extends the theory of functional principal component analysis to regularly varying functions, removing the need for fourth moment assumptions and establishing new asymptotic properties and consistency results.

## Contribution

It develops asymptotic results for PCA of functional data under regular variation, a setting without the fourth moment requirement, and applies these to functional linear models.

## Key findings

- Derived the asymptotic distribution of sample covariance operators
- Established convergence of moments and almost sure convergence
- Proved the consistency of the regression operator in functional linear models

## Abstract

The paper is concerned with asymptotic properties of the principal components analysis of functional data. The currently available results assume the existence of the fourth moment. We develop analogous results in a setting which does not require this assumption. Instead, we assume that the observed functions are regularly varying. We derive the asymptotic distribution of the sample covariance operator and of the sample functional principal components. We obtain a number of results on the convergence of moments and almost sure convergence. We apply the new theory to establish the consistency of the regression operator in a functional linear model.