Random Matrix Theory [/ˈrændəm ˈmeɪtrɪks ˈθɪri/] n - Random matrix theory tells what the eigenvalues of a covariance matrix should look like when the data is mere noise. If your sample eigenvalues answer to that prediction, you are fitting noise. If a few stand out above the bulk, they may carry tidings.

For an $N\times T$ matrix of independent, identically distributed returns with variance ${\sigma }^2$, the Marchenko-Pastur law gives the limiting distribution of the eigenvalues. Most such as spring from noise fall within a predictable bulk:

Random matrix eigenvalues versus Marchenko-Pastur

The support of that bulk is:

$${\lambda }_{\pm }={\sigma }^2{\left(1\pm \sqrt{\frac{N}{T}}\right)}^2$$

When $N$ is large against $T$, the upper edge can reach surprisingly high. A covariance matrix reckoned from fifty assets and a hundred days will show eigenvalues that look meaningful, though no true structure underlies them.

In practice one cleans the matrix by setting the eigenvalues within the Marchenko-Pastur band to their average, or by shrinking them toward a target. What results is a stabler estimate for Modern Portfolio Optimization and Principal Components and Eigenportfolios.

The theory serves best when $N$ and $T$ are comparable. Ten thousand observations of ten assets, and noise is not your master; two hundred of a hundred, and it is. The ratio $N/T$ is the thing to watch.

It assumes independent, identically distributed returns of finite variance. Real returns are none of these, and their tails run fatter than the assumptions allow; the threshold is a heuristic, not a theorem to invoke blind. It stands as one more reason to mistrust a covariance matrix built upon too little data.