# sampling distribution of covariance of two normal distribution

Let $X$, $Y$ be two independent normal random variable, say $N(0,1)$. And I want to estimate $cov(X,Y)$. What is the distribution of the $\widehat{cov}(X,Y)$?

• Of course, I can calculate the true value directly with $cov(X,Y)=0$.
• Suppose I have data $X_1,\cdots,X_n \sim N(0,1)$ and $Y_1,\cdots,Y_n \sim (another\ independent) N(0,1)$
• Consider the estimator $\widehat{cov}(X,Y)=\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})$

Then my question is what is the distribution of $\widehat{cov}(X,Y)$?

• I did a simulation and the result is very similar to a normal distribution. Is is possible to prove that it is normal, with, like, CLT? – breezeintopl Apr 4 '17 at 13:45
• The normal distribution is about $N(0,(\frac{1}{\sqrt{n}})^2)$. – breezeintopl Apr 4 '17 at 14:00

The answer seems to be given here (see $f(x_{12})$). Although I can't see the reference.
It's not quite as ugly as it seems -- it resembles the chi-square distribution, but is also modulated by a Bessel function. Note the simplification when $\rho = 0$, as in your question.