The Product of Gaussian Matrices is Close to Gaussian

We study the distribution of the it matrix product G1 G2 cdots Gr of r independent Gaussian matrices of various sizes, where Gi is di-1 × di, and we denote p = d0, q = dr, and require d1 = dr-1. Here the entries in each Gi are standard normal random variables with mean 0 and variance 1. Such products arise in the study of wireless communication, dynamical systems, and quantum transport, among other places. We show that, provided each di, i = 1, ldots, r, satisfies di ≥ C p · q, where C ≥ C0 for a constant C0 > 0 depending on r, then the matrix product G1 G2 cdots Gr has variation distance at most δ to a p × q matrix G of i.i.d. standard normal random variables with mean 0 and variance prodi=1r-1 di. Here δ rightarrow 0 as C rightarrow ∞. Moreover, we show a converse for constant r that if di < C' maxp,q1/2minp,q3/2 for some i, then this total variation distance is at least δ', for an absolute constant δ' > 0 depending on C' and r. This converse is best possible when p=Theta(q).