distribution of sample variance of correlated observations

Question

It is well known that if we have n i.i.d. observations of a normal random variable, then Cochran's theorem tells us that:

$\frac{(n-1)s^2}{\sigma^2} \widetilde{} χ^2_{n-1}$

But what if the samples are not i.i.d., but they are correlated? Is there any expression or theorem that gives us an insight on the distribution of the sample variance, assuming that we know a-priori the correlation matrix between the samples?

For example assume that the samples are extracted from a realization of a stationary gaussian random field and that the correlation function $\rho(\tau)$ is completely known.

Answer 1

Of course if you know the correlation matrix you can derive an expression for sample variance distribution - but I think it will not be a closed form distribution.

Let $X$ be your sample. You can express it as $X = AY+m$, where Y is iid standard normal, $A$ is a matrix and $m$ is deterministic mean.

In such a setting a variable of interest is $$\sum{(X_i-\bar{X})^2}=||X-\bar{X}||^2=||(AY+m-\bar{A}Y-m)||^2=||(A-\bar{A})Y||^2$$

Mean $m$ cancels out. You finally get a weighted sum of squares of standard normal variable (note that weights are not generally equal, so it is not a chi-squared distribution) and possibly some cross-products. If you know your correlation matrix you may derive matrix A (eg. Cholesky decomposition) and find the parameters, but I am not sure if this is useful for practical purpose.