ChapterZero

It turns out that the question I’m looking at now is strongly connected the question of what difference there is between the raw and central matrix moments of random self-adjoint matrices. Remember the raw moments of a centered scalar random variable are while the central moments are . The same definitions hold in the matrix [...]

This post is a brief overview of the technique introduced in Joel’s User Friendly technical report (the User Friendly preprint is more intricate, but also more powerful, because it generalizes the results in the technical report to the case of martingales). I realize I’ve mentioned this paper in passing in several posts. I’m pretty obsessed [...]

Let be strictly positive integers. I’d like to know that Any ideas? It looks like the integrand is pretty flat near 0, then it dips below zero, then shoots up above zero again strongly enough that the integral is always positive. The integral as a function of also seems to be decreasing. Equivalently, I’d like [...]

I found this result while trying to prove that the moments of a sum of scaled chi-squared random variables grow like : Let be a standard Gaussian random variable. Then for any positive integers (I may have missed some way in which this is trivially true anyhow, but I don’t think so) (Update: in the [...]

Here’s an interesting proof of the CS-ineq that uses Schur Complements. Let be square-integrable functions. Note that , so the expectation of this matrix is positive. This implies the Schur complement is positive. This is the Cauchy-Schwarz inequality. I mention this because exactly the same trick gives you a matrix-valued Cauchy-Schwarz inequality: Possibly relevant posts: [...]

Ok, I’ll put this out there. I’m try to calculate the matrix moments of a Wishart matrix with one degree of freedom: I have and What is in terms of ? I was thinking of applying Isserlis’ theorem and the relation but good God I hate dealing with the combinatorics that would involve, and I [...]

I’m thinking about how to measure the convergence of a sample covariance matrix to the covariance matrix. To be specific, let , then we have by the law of large numbers. Both the sample covariance matrix and the covariance matrices are positive semidefinite, so we can bound the error between the spectra of the two [...]

Say I have a sum of random Hermitian matrices . I’d like to know what the -th eigenvalue of tends to look like. In the case that the are positive-semidefinite, the upper and lower Chernoff tail bounds in User Friendly Tail Bounds bound the relative deviation of from and from respectively, so you might expect [...]

Let be conformal matrices, then it’s clear that the matrix is positive semidefinite, because if you consider as vectors, it is their Grammian. It is not true in general that for a fixed positive integer is positive semidefinite. However, it might be true that if we add these matrices for each together that the result [...]