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

Abstract

Let $(X_i,i\ge 1)$ 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 $t\mapsto {\sum }_{i=1}^{n}f(X_i)1_{X_i\le t}$.

9 pages