The Product of Gaussian Matrices is Close to Gaussian
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We study the distribution of the matrix product G_1 G_2 ⋯ G_r of r independent Gaussian matrices of various sizes, where G_i is d_i-1× d_i, and we denote p = d_0, q = d_r, and require d_1 = d_r-1. Here the entries in each G_i 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 d_i, i = 1, …, r, satisfies d_i ≥ C p · q, where C ≥ C_0 for a constant C_0 > 0 depending on r, then the matrix product G_1 G_2 ⋯ G_r 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 ∏_i=1^r-1 d_i. Here δ→ 0 as C →∞. Moreover, we show a converse for constant r that if d_i < C' max{p,q}^1/2min{p,q}^3/2 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=Θ(q).
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