A matrix concentration inequality for products

08/12/2020

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by   Sina Baghal, et al.

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We present a non-asymptotic concentration inequality for the random matrix product Z_n = (I_d-α X_n)(I_d-α X_n-1)⋯(I_d-α X_1), where {X_k }_k=1^+∞ is a sequence of bounded independent random positive semidefinite matrices with common expectation 𝔼[X_k]=Σ. Under these assumptions, we show that, for small enough positive α, Z_n satisfies the concentration inequality ℙ(‖ Z_n-𝔼[Z_n]‖≥ t) ≤ 2d^2·(-t^2/ασ^2) for all t≥ 0, where σ^2 denotes a variance parameter.