Co-variance Operator of Banach Valued Random Elements: U-Statistic Approach

TL;DR

This paper introduces a covariance operator for Banach space-valued random elements using U-statistics, analyzing its asymptotic behavior and extending results to dependent data in Hilbert spaces.

Contribution

It develops a novel covariance operator framework for Banach and Hilbert space-valued random elements based on U-statistics, including asymptotic distribution results.

Findings

Asymptotic distribution derived for the covariance operator in Banach spaces.

Extension of results to dependent data under mixing conditions.

Provides large sample properties of the proposed estimator.

Abstract

This article proposes a co-variance operator for Banach valued random elements using the concept of $U$-statistic. We then study the asymptotic distribution of the proposed co-variance operator along with related large sample properties. Moreover, specifically for Hilbert space valued random elements, the asymptotic distribution of the proposed estimator is derived even for dependent data under some mixing conditions.