I've written a paper for university on Random Matrices and during my research I've had an interesting idea, let me explain: Wigner's Semicircle Law has seen much advancement since its original proof in 1955, most recently I believe being Tao's proof of the Wigner-Gaudin-Mehta-Dyson conjecture showing universality. Now here's the leap, much of big data is reliant on Principle Component Analysis, or decomposing data into their respective eigenvalues and eigenvectors. Then we compare the results to similar datasets to see if there are correlations. However if we treat stock prices as Brownian motions, i.e. iterated random processes with eigenvalues and eigenvectors tending to the circular law, then doesn't that inherently create a bias in our comparison of the eigenvectors w.r.t other iterated random processes.
For example a group of commodities stocks in agriculture and another in mining we assume aren't correlated, but after batching and PCA they share similar normed eigenvalues. Is this not in part due to the fact they share the same distributive law at least for large enough batches and repetitive samplings? If so are there already methods or hypothesis tests that filter through this?
It was just a thought and I don't really have many people to discuss this idea with seeing as I'm stuck at home. I may be wrong on how PCA works or how financial products are correlated as I'm not in the field.
EDIT: I feel like some further context is needed since this isn't a result familiar to most.
From RMT, eigenvalues have a semicircle distribution for symmetric matrices each with i.i.d normally distributed entries. The restrictions on i.i.d have recently been shown to not matter so we can proceed nevertheless. If we take a covariance matrix of all stock tickets beginning in A comparing the average daily return over a period of time, each we can assume lognormal distribution forming a, lets say 10000 by 10000 symmetric matrix. Thus we get a sequence of random covariance matrices $\Gamma_1, \Gamma_2, ..., \Gamma_n$. Each of the entries we assume to be i.i.d since the stocks have 'nothing' to do with each other (although a weaker results holds for non-i.i.d entries). Now this series of matrices forms a chain of covariance matrices tending towards the underlying covariance matrix of the entire history of the stocks (if we sampled correctly). We known from RMT that once we decompose these matrices into their eigenvalues, the eigenvalues tend to the semicircle distribution. Since this distribution is continuous, there's a spread in results i.e. there is some underlying variance to the eigenvalue decomposition of covariance matrices. Thus when we use covariance matrices shouldn't there be some sort of hypothesis test that's able to filter out this underlying distribution, similar to comparing normal distributions where we need to account for variance when comparing two mean values. This would be dependent on how i.i.d the random variables are, the size of the matrix, the number of samples taken, and the mean/variance of the random variables themselves. What would be weird about this hypothesis test is that we'd expects as $n$ gets larger so does the error bound, capture by the asymptotic relationship between the size and convergence to the semicircle distribution.
TLDR: Is there some sort of hypothesis testing for PCA, or any eigenvalue method, that filters out the underlying tendency of random covariance matrices so as to account for the variance? Similar to how when you compare the mean of two normal distributions you need to perform a hypothesis test to account for the variance.
Also the more I write about this, the more I feel as though this is more related to data science as opposed to quantative finance as I realise my examples don't seem to fit very well.