2018
DOI: 10.3150/17-bej961
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Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models

Abstract: Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results for random graphs, germ-grain models in stochastic geometry and multinomial allocation models. The results obtained compare favorably with classical methods, including the use of McDiarmid's inequality, negative association, and self bounding functions.

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Cited by 9 publications
(25 citation statements)
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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%
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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%
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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%
“…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%