For a sample of observations $\{x_i\}_{i=1}^n$ where $x_i=(x_{i1},\dots,x_{ik})^T$ of a population random vector $X=(X_1,\dots,X_k)^T$, the population covariance is $$ \Sigma = E[(X-E[X])(X-E[X])^T], $$ and the sample covariance is $$ \hat \Sigma = \frac{1}{n} \sum_{i=1}^n (x_i-\bar{x})(x_i-\bar{x})^T. $$
Now, while $\hat \Sigma$ is an unbiased estimator of $\Sigma$, the same is not true for the inverse. If the samples are drawn from a multivariate normal distribution it can be shown that $$ E[\hat \Sigma^{-1}] = \frac{n}{n-k-2}\Sigma^{-1}. $$ So the sample covariance is biased, but at least as $n$ gets large the bias will decrease and thus $$ \lim_{n \to \infty }E[\hat \Sigma^{-1}] \to \Sigma^{-1}. $$
Now, what about the case when the samples are drawn from a different continuous distribution. Is there anything concrete we can say about $\hat \Sigma^{-1}$ as an estimator of $\Sigma^{-1}$?
In particular, will $E[\hat \Sigma^{-1}] \to \Sigma^{-1}$ still hold? If it doesn't hold in general, for what class of random variables does it still hold? Or does it really only hold in the case of a multivariate normal distribution?
