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User-Friendly Tail Bounds for Sums of Random Matrices


This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales.

In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.


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Correspondence to Joel A. Tropp.

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Communicated by Albert Cohen.

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Tropp, J.A. User-Friendly Tail Bounds for Sums of Random Matrices. Found Comput Math 12, 389–434 (2012).

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  • Discrete-time martingale
  • Large deviation
  • Probability inequality
  • Random matrix
  • Sum of independent random variables

Mathematics Subject Classification (2000)

  • 60B20
  • 60F10
  • 60G50
  • 60G42