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On some extensions of Bernstein’s inequality for self-adjoint operators

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  • Minsker, Stanislav

Abstract

We present some extensions of Bernstein’s concentration inequality for random matrices. This inequality has become a useful and powerful tool for many problems in statistics, signal processing and theoretical computer science. The main feature of our bounds is that, unlike the majority of previous related results, they do not depend on the dimension d of the ambient space. Instead, the dimension factor is replaced by the “effective rank” associated with the underlying distribution that is bounded from above by d. In particular, this makes an extension to the infinite-dimensional setting possible. Our inequalities refine earlier results in this direction obtained by Hsu et al. (2012).

Suggested Citation

  • Minsker, Stanislav, 2017. "On some extensions of Bernstein’s inequality for self-adjoint operators," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 111-119.
    • Handle: RePEc:eee:stapro:v:127:y:2017:i:c:p:111-119
    DOI: 10.1016/j.spl.2017.03.020
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    References listed on IDEAS

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    1. Ahmad, I.A. & Amezziane, M., 2013. "Probability inequalities for bounded random vectors," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1136-1142.
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    Cited by:

    1. Bernhard G. Bodmann & Martin Ehler & Manuel Gräf, 2018. "From Low- to High-Dimensional Moments Without Magic," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2167-2193, December.
    2. Dong Xia & Ming Yuan, 2021. "Statistical inferences of linear forms for noisy matrix completion," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 58-77, February.
    3. Kim Christensen & Mikkel Slot Nielsen & Mark Podolskij, 2023. "High-dimensional estimation of quadratic variation based on penalized realized variance," Statistical Inference for Stochastic Processes, Springer, vol. 26(2), pages 331-359, July.

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