# Marchenko–Pastur, Student distribution and returns

I have a question regarding random matrix theory. I've been studying various papers and I found some confusing definitions of Marchenko-Pastur law. The most clear was the one on wiki: wiki-Pastur-Marchenko

However I've checked quite a few papers on connection between financial data, RMT and covariance matrix of returns. I found them to be very confusing. In particluar I've been reading in detail this paper (I would have posted more links but can't): paper1. As I understand (and wiki also says so) Marchenko-Pastur law holds for matrices H such that: $H=\frac{1}{N}(XX^T)$, where X is $M\times N$ matrix whose entries have zero mean and finite second moment.

The paper linked above mentions returns have multivariate Student distribution, are correlated and they go deeper into dissecting the distribution. And than they go on and develop something similar to Marchenko–Pastur law. Shouldn't Marchenko-Pastur also hold for multivariate Student distribution? Or it does not hold because not all entries are distributed by the same marginal distribution? In particluar, I'm interesting in further explanation of eqs. 7 and 8. I think in case when all marginal distributions of multivariate Student are the same, it should converge to Marchenko-Pastur, however I just don't see this happening.

Thanks for help.

• I figured it out. The definition of returns in the paper is such that they are not independent any more. If one looks at the paper, one can see that in definition of return there is common $\sigma_t$ factor that makes Marchenko–Pastur invalid. However, if one uses $P(s)=\delta(s-1)$ one recovers Marchenko–Pastur. – Jur Mar 23 '16 at 16:57