What is the distribution of empirical covariance between two independent normal distributions?

Question

Suppose that we have two independent normal distributions $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ what is the distribution of empirical covariance (or empirical correlation if this make my question simpler ) between two samples of (fixed) size $k$ ( observations are indexed by ${t}$ ) sampled from $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ ?
Simply approach involving convolution seems to doesn't work for me well, because $2k-1$ dimensional integral appears with which I can't deal.

I've made some simulations for $k=10,9,8,...,2$ we have densities :

here is simple R code :

cov_<-function(k){
   cov(rnorm(k),rnorm(k))
}

sim<-function(n,ile){
   wyn<-numeric(n)
   for(i in 1:n){
      wyn[i]=cov_(ile)
   }
   wyn
}



plot(density(sim(10000,10)))

for(j in 9:2){
    lines(density(sim(10000,j)),col=j-1)
}

Answer 1

The empirical covariance converges to the theoretical covariance (in your case, zero) at the rate $1/\sqrt{k}$ when $k\to\infty$ but it has no simple distribution for fixed finite $k$. See the first pages of this paper to get an idea of the situation.