# Weak convergence of an empirical process. Demonstration

The empirical process $B_n(x) = \sqrt{n}(F_n(x) − F(x))$ converges weakly to a zero-mean Gaussian process, $B$, with covariance function:

$\mbox{cov}(B(x), B(y)) = F(\min\{x, y\}) − F(x)F(y)$.

How I can prove this assumption (about covariance function)?

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Do you know some basics notions of the theory of empirical processes ? – Stéphane Laurent Oct 14 '12 at 19:33
Yes, I know. Why do you ask this? – anxoestevez Oct 14 '12 at 19:40
To put your question in a context. The question you ask is proved in any introductory textbook about empirical processes. – Stéphane Laurent Oct 14 '12 at 20:00
Would you recommend me an introductory textbook? Thanks! – anxoestevez Oct 14 '12 at 21:17

This result is called Donsker's theorem. It is proved in any textbook about the theory of empirical processes, such as:

You should also easily find some courses about empirical processes with the help of Google.

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