Exponential bounds for the hypergeometric distribution

Greene, Evan; Wellner, Jon A.

doi:10.3150/15-bej800

Cited by 19 publications

(24 citation statements)

References 26 publications

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“…However, the estimation protocol does essentially nothing but determine the two error rates e x and e z . The expected values of these rates can be readily obtained from equation (13). The error rate e z vanishes, because dephasing in the Z-basis leaves the Z-diagonal invariant.…”

Section: Discussionmentioning

confidence: 99%

Capacity estimation and verification of quantum channels with arbitrarily correlated errors

et al. 2018

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The central figure of merit for quantum memories and quantum communication devices is their capacity to store and transmit quantum information. Here, we present a protocol that estimates a lower bound on a channel’s quantum capacity, even when there are arbitrarily correlated errors. One application of these protocols is to test the performance of quantum repeaters for transmitting quantum information. Our protocol is easy to implement and comes in two versions. The first estimates the one-shot quantum capacity by preparing and measuring in two different bases, where all involved qubits are used as test qubits. The second verifies on-the-fly that a channel’s one-shot quantum capacity exceeds a minimal tolerated value while storing or communicating data. We discuss the performance using simple examples, such as the dephasing channel for which our method is asymptotically optimal. Finally, we apply our method to a superconducting qubit in experiment.

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Section: Discussionmentioning

confidence: 99%

Capacity estimation and verification of quantum channels with arbitrarily correlated errors

et al. 2018

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“…One of the most famous concentration results for sampling without replacement is Serfling's inequality [27], which can be regarded as a strengthening of Hoeffding's inequality for n out of N sampling due to the inclusion of the finite-sampling correction factor 1 − n/N . For a discussion and some newer results, we refer to [6,19,30].…”

Section: Proposition 2 Let F : ω κN → R Be An Arbitrary Function Andmentioning

confidence: 99%

“…In [19], it was conjectured that √ n f has sub-Gaussian tails with variance 1 − n/N . The next result states that after centering around the expectation, this is indeed the case.…”

Section: Proposition 2 Let F : ω κN → R Be An Arbitrary Function Andmentioning

confidence: 99%

Concentration Inequalities on the Multislice and for Sampling Without Replacement

Sambale

Sinulis

2021

J Theor Probab

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We present concentration inequalities on the multislice which are based on (modified) log-Sobolev inequalities. This includes bounds for convex functions and multilinear polynomials. As an application, we show concentration results for the triangle count in the G(n, M) Erdős–Rényi model resembling known bounds in the G(n, p) case. Moreover, we give a proof of Talagrand’s convex distance inequality for the multislice. Interpreting the multislice in a sampling without replacement context, we furthermore present concentration results for n out of N sampling without replacement. Based on a bounded difference inequality involving the finite-sampling correction factor $$1 - (n / N)$$ 1 - ( n / N ) , we present an easy proof of Serfling’s inequality with a slightly worse factor in the exponent, as well as a sub-Gaussian right tail for the Kolmogorov distance between the empirical measure and the true distribution of the sample.

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“…The proof of this bound, along with a complete analogue for the hypergeometric distribution of a bound of Talagrand (1994) for the binomial distribution, appears in Greene and Wellner (2015) and in the forthcoming Ph.D. thesis of the first author, Greene (2016).…”

Section: Introduction: Serfling’s Finite Sampling Exponential Boundmentioning

confidence: 99%

“…(Greene and Wellner (2015); Greene (2016)) Suppose that

\sum_{i = 1}^{n} Y_{i} ~ Hypergeometric (n, D, N)

. Define μ N = D / N and suppose N > 4 and 2 ≤ n < D ≤ N /2.…”

Section: Introduction: Serfling’s Finite Sampling Exponential Boundmentioning

confidence: 99%

Finite sampling inequalities: An application to two-sample Kolmogorov–Smirnov statistics

Greene

Wellner

2016

Stochastic Processes and their Applications

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We review a finite-sampling exponential bound due to Serfling and discuss related exponential bounds for the hypergeometric distribution. We then discuss how such bounds motivate some new results for two-sample empirical processes. Our development complements recent results by Wei and Dudley (2012) concerning exponential bounds for two-sided Kolmogorov - Smirnov statistics by giving corresponding results for one-sided statistics with emphasis on “adjusted” inequalities of the type proved originally by Dvoretzky et al. (1956) and by Massart (1990) for one-sample versions of these statistics.

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Exponential bounds for the hypergeometric distribution

Cited by 19 publications

References 26 publications

Capacity estimation and verification of quantum channels with arbitrarily correlated errors

Capacity estimation and verification of quantum channels with arbitrarily correlated errors

Concentration Inequalities on the Multislice and for Sampling Without Replacement

Finite sampling inequalities: An application to two-sample Kolmogorov–Smirnov statistics

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