I'm trying to figure out if Wishart can be close to Normal for a number of degrees of freedom enough large.

About chi-squared distribution, Wikipedia states:

By the central limit theorem, because the chi-squared distribution is the sum of k 
independent random variables with finite mean and variance, it converges to a normal 
distribution for large k. For many practical purposes, for k > 50 the distribution is 
sufficiently close to a normal distribution for the difference to be ignored.

Since Wishart is presented as a generalization to multiple dimensions of the chi-squared distribution, is there a similar property?


The Wishart distribution (when dimension is 2 or higher) is a distribution on a space of matrices, if the underlying space had dimension $d$, the space of symmetric positive matrices $d \times d$ has dimension $d(d+1)/2$, so for approximating Wishart by a normal you would need a normal of that dimension which in addition fulfills the (non-linear) conditions necessary to be a positive definite matrix. That doesn't sound like being possible, so I believe the answer to your question is NO.


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