Asymptotics of the Empirical Cross-over Function

TL;DR

This paper introduces the Empirical Cross-over Function, an L-statistic related to clustering criteria, and establishes its asymptotic properties including law of large numbers, CLT, and functional CLT.

Contribution

It provides the first limit theorems for the Empirical Cross-over Function, a novel L-statistic with irregular weights, expanding understanding of its asymptotic behavior.

Findings

Proved law of large numbers for the Empirical Cross-over Function.

Established central limit theorem for the function.

Derived a functional CLT demonstrating its convergence in distribution.

Abstract

We consider a combination of heavily trimmed sums and sample quantiles which arises when examining properties of clustering criteria and prove limit theorems. The object of interest, which we call the Empirical Cross-over Function, is an L-statistic whose weights do not comply with the requisite regularity conditions for usage of ex- isting limit results. The law of large numbers, CLT and a functional CLT are proven.