Covariance of binomial variables

Question

I can't demonstrate that the covariance of G is like in (3): I can't understand because there's that factor "n". Can someone help me please? Thank you!

Answer 1

Just write $G_n(t)$ and $G_n(s)$ as sum of those indicators .

So the product is $n^2G_n(t)G_n(s)=\Sigma_{i,j=1}^{\infty}1_{\xi_i<t}1_{\xi_j<t}$ Now if $i=j$ then $1_{\xi_i<t}1_{\xi_j<t}=1_{\xi_i<min(s,t)}$.If $i \neq j$ then they are independent.

else $n^2G_n(t)G_n(s)=\Sigma_{i\neq j}^{\infty}1_{\xi_i<t}1_{\xi_j<t} + \Sigma_{i,1}^{\infty}1_{\xi_i<min(s,t)}$

Taking expectation you get.

$E(G_n(t)G_n(s))=\frac{1}{n^2}(n(min(s,t))+n(n-1)st)$

so $Cov(G_n(t),G_n(s))=\frac{1}{n^2}(n(min(s,t)+n(n-1)st)-st \\=\frac{1}{n^2}(n(min(s,t))-nst) \\ \implies nCov(G_n(t),G_n(s))=min(s,t)-st$