I was wondering if there is a common solution for the covariance of a random variable $x[n]$ (i.i.d) and its mean estimator $\hat{\theta}$. Mathematically,
\begin{equation} \sigma^2 \left ( x[n],\hat{\theta} \right ) \end{equation}
Where,
\begin{equation} \hat{\theta} = \frac{1}{N}\sum_{n=0}^{N-1}x[n] \end{equation}
Thank you.