Please help me proving this:

Suppose that $Y_1,\ldots,Y_n$ are i.i.d. nonnegative RV's with CDF $F$ and $E(Y_i)=\mu<\infty$. Let $y_1,\ldots,y_n$ be a realization from which an EDF $\hat{F}$ is defined: $$ \hat{F}(y)=\frac{1}{n}\sum_{i=1}^nI(y_i\leq y). $$ Want to show: $$ \text{plim}_{n\to\infty}\left(n^{1/2}\int_0^\infty y(\hat{F}(y)-F(y))d(\hat{F}-F)(y)\right)=0. $$ Please state any further necessary assumptions.

EDIT responding to GUNG'S comment:

This result is a small step in a bigger argument in Russell Davidson 2009's paper on the Gini index. The author must have considered it obvious because he mentions it without proof.

I have some understanding of asymptotic theory such as some laws of large numbers, some variants of the CLT, various modes of convergence, Slutsky Theorem, measure theory in general and probability theory in particular, and some stochastic calculus.

My problem is I've been learning asymptotic theory on my own and I haven't done a lot of exercises in this. So while I want to understand the author's reasoning, I don't know where to start with this one.

  • $\begingroup$ We welcome questions like this, @user41277, but we treat them differently. Please tell us what you understand thus far, what you've tried & where you are stuck, & we'll try to provide hints to get you unstuck. To better understand the process, you should read the wiki for the [self-study] tag. Please edit your question to add this information, as simply listing your homework question & hoping someone will provide an answer for you is grounds for closing. $\endgroup$ Mar 4, 2014 at 0:45
  • $\begingroup$ I suggest reading up about convergence of empirical processes. I would start with Glivenko-Cantelli and Donsker type theorems. Good books for that are written by A. van der Vaart, Assymptotic Statistics and also Empirical Processes. $\endgroup$
    – mpiktas
    Mar 4, 2014 at 8:04
  • $\begingroup$ Hi mpiktas, thank you for pointing me to the van der Vaart book. It actually helped: my original problem was to justify an asymptotic approximation $A\approx B$ and, with the information in Asymptotic Statistics, I was able to do that (informally) with von Mises calculus. But I'm still very interested in learning how one would prove the assertion in my original post. I studied some variants of Glivenko-Cantelli and Donsker theorems but I'm still stuck. Thanks! $\endgroup$ Mar 17, 2014 at 14:05


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