Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models

Bartroff, Jay; Goldstein, Larry; Işlak, Ümi̇t

doi:10.3150/17-bej961

Cited by 9 publications

(25 citation statements)

References 59 publications

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“…Size bias also plays a role in concentration inequalities, see [40,39,8,16]. The results from [40,8] include: if X ≥ 0 with a := EX ∈ (0, ∞) can be coupled to X * so that P(X * ≤ X + c) = 1, then…”

Section: Relation To Stein's Methods and Concentration Inequalitiesmentioning

confidence: 99%

“…The condition P(X * > t) ≥ P(X > t) for all t is described as "X * lies above X in distribution," written X ≤ st X * , and implies that there exist couplings of X * and X in which always X ≤ X * . Writing Y for the difference, we have (16) X…”

Section: Stochastic Monotonicitymentioning

confidence: 99%

“…Another issue is whether the lognormal can be expressed as a mixture of these discrete distributions. Leipnik 1991, [60, page 337], wrote 16 "One hopes that for some mixing distribution dh(b) we have that the lognormal distribution for e σZ is a mixture, governed by h, of the single orbit distributions, and so too…”

Section: Size Bias the Lognormal And Chihara-leipnikmentioning

confidence: 99%

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Size bias for one and all

Arratia¹,

Goldstein²,

Kochman³

2019

Probab. Surveys

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Section: Relation To Stein's Methods and Concentration Inequalitiesmentioning

confidence: 99%

Section: Stochastic Monotonicitymentioning

confidence: 99%

Section: Size Bias the Lognormal And Chihara-leipnikmentioning

confidence: 99%

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Size bias for one and all

Arratia¹,

Goldstein²,

Kochman³

2019

Probab. Surveys

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“…Hence, for sums of independent non-negative variables whose expectation is small in comparison to their L ∞ norms, Theorem 3.3 provides better estimates than Azuma-Hoeffdingtype (or "bounded differences") inequalities, such as [Hoe63, line (2.3)] (the bound that is widely referred to as "Hoeffding's inequality"). See also Section 4 of [BGI14] for a fuller comparison of concentration results obtained by bounded size bias couplings to those via more classical means.…”

Section: Example 32 (Size Biased Couplings For Independent Sums) Sumentioning

confidence: 99%

Size biased couplings and the spectral gap for random regular graphs

Cook¹,

Goldstein²,

Johnson³

2018

Ann. Probab.

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Let λ be the second largest eigenvalue in absolute value of a uniform random dregular graph on n vertices. It was famously conjectured by Alon and proved by Friedman that if d is fixed independent of n, then λ = 2 √ d − 1 + o(1) with high probability. In the present work we show that λ = O( √ d) continues to hold with high probability as long as d = O(n 2/3 ), making progress towards a conjecture of Vu that the bound holds for all 1 ≤ d ≤ n/2. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at d = o(n 1/2 ). We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on d-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.

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“…It is worth noting that the variance bound EK n,r in this concentration inequality can also be obtained using a variant of Stein's method known as size-biased coupling (Bartroff et al [5], Chen et al [15]).…”

Section: Concentration Inequalitiesmentioning

confidence: 99%

Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications

Ben-Hamou¹,

Boucheron²,

Ohannessian³

2017

Bernoulli

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An infinite urn scheme is defined by a probability mass function (pj) j≥1 over positive integers. A random allocation consists of a sample of N independent drawings according to this probability distribution where N may be deterministic or Poisson-distributed. This paper is concerned with occupancy counts, that is with the number of symbols with r or at least r occurrences in the sample, and with the missing mass that is the total probability of all symbols that do not occur in the sample. Without any further assumption on the sampling distribution, these random quantities are shown to satisfy Bernstein-type concentration inequalities. The variance factors in these concentration inequalities are shown to be tight if the sampling distribution satisfies a regular variation property. This regular variation property reads as follows. Let the number of symbols with probability larger than x be ν(x) = |{j: pj ≥ x}|. In a regularly varying urn scheme, ν satisfies limτ→0 ν(τ x)/ ν(τ ) = x −α for α ∈ [0, 1] and the variance of the number of distinct symbols in a sample tends to infinity as the sample size tends to infinity. Among other applications, these concentration inequalities allow us to derive tight confidence intervals for the Good-Turing estimator of the missing mass.

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Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models

Cited by 9 publications

References 59 publications

Size bias for one and all

Size bias for one and all

Size biased couplings and the spectral gap for random regular graphs

Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications

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