A functional central limit theorem for the partial sums of sorted i.i.d. random variables

TL;DR

This paper establishes a functional central limit theorem for the partial sums of sorted i.i.d. random variables transformed by a function, providing a new understanding of their asymptotic behavior.

Contribution

It introduces a functional CLT for the process involving sums of functions of sorted i.i.d. variables, extending classical results to this setting.

Findings

Proves a functional CLT for the process involving sorted i.i.d. variables.

Shows convergence of the process to a Gaussian process.

Extends the theory of empirical processes with a new functional limit theorem.

Abstract

Let $(X_i,i\geq 1)$ be a sequence of i.i.d. random variables with values in $[0,1]$, and $f$ be a function such that $`E(f(X_1)^2)<+\infty$. We show a functional central limit theorem for the process $t\mapsto \sum_{i=1}^n f(X_i)1_{X_i\leq t}$.