Weak convergence of an empirical process. Demonstration

Question

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)?

Answer 1

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

• Kosorok's book "Introduction to Empirical Processes and Semiparametric Inference" (a pdf version is available here)

• van der Vaart and Wellner's book "Weak Convergence and Empirical Processes"

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