Extension of Donsker's Invariance Principle with Incomplete Partial-Sum Process

TL;DR

This paper extends Donsker's invariance principle to incomplete partial-sum processes of i.i.d. random variables, broadening the scope of functional and deleting-item central limit theories.

Contribution

It introduces new invariance principles for incomplete partial-sum processes, expanding the theoretical framework of weak convergence and central limit theorems.

Findings

Extended Donsker's invariance principles to incomplete processes

Established invariance principles for empirical processes

Enriched the structure of weak convergence in probability theory

Abstract

Based on deleting-item central limit theory, the classical Donsker's theorem of partial-sum process of independent and identically distributed (i.i.d.) random variables is extended to incomplete partial-sum process. The incomplete partial-sum process Donsker's invariance principles are constructed and derived for general partial-sum process of i.i.d random variables and empirical process respectively, they are not only the extension of functional central limit theory, but also the extension of deleting-item central limit theory. Our work enriches the random elements structure of weak convergence.