SCHWEITZER 1983 Problem 12

Let $X_1,X_2,\ldots, X_n$ be independent, identically distributed, nonnegative random variables with a common continuous distribution function $F$ . Suppose in addition that the inverse of $F$ , the quantile function $Q$ , is also continuous and $Q(0)=0$ . Let $0=X_{0: n} \leq X_{1: n} \leq \ldots \leq X_{n: n}$ be the ordered sample from the above random variables. Prove that if $EX_1$ is finite, then the random variable $$\Delta = \sup_{0\leq y \leq 1} \left| \frac 1n \sum_{i=1}^{\lfloor ny \rfloor +1} (n+1-i)(X_{i: n}-X_{i-1: n})- \int_0^y (1-u)dQ(u) \right|$$ tends to zero with probability one as $n \rightarrow \infty$ .

S. Csorgp, L. Horvath