Noncommutative Bennett and Rosenthal inequalities. (English) Zbl 1290.46056
Summary: In this paper, we extend the Bernstein, Prohorov and Bennett inequalities to the noncommutative setting. In addition, we provide an improved version of the noncommutative Rosenthal inequality, essentially due to S. V. Nagaev and I. F. Pinelis [Theory Probab. Appl. 22(1977), No. 2, 254–263 (1978); translation from Teor. Veroyatn. Primen. 22, 254–263 (1977; Zbl 0378.60036)] and I. F. Pinelis and S. A. Utev [Theory Probab. Appl. 29, 574–577 (1985); translation from Teor. Veroyatn. Primen. 29, No. 3, 554–557 (1984; Zbl 0566.60017)] for commutative random variables. We also present new best constants in Rosenthal’s inequality. Applying these results to random Fourier projections, we recover and elaborate on fundamental results from compressed sensing, due to E. J. Candès, J. K. Romberg and T. Tao [IEEE Trans. Inf. Theory 52, No. 2, 489–509 (2006; Zbl 1231.94017)].
MSC:
| 46L53 | Noncommutative probability and statistics |
| 60E15 | Inequalities; stochastic orderings |
| 46L52 | Noncommutative function spaces |
| 60F10 | Large deviations |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
Keywords:
(noncommutative) Bennett inequality; (noncommutative) Rosenthal inequality; (noncommutative) Bernstein inequality; (noncommutative) Prohorov inequality; noncommutative \(L_{p}\) spaces; compressed sensing; large deviation; Cramér’s theoremReferences:
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