Berry-Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing

Goldstein, Larry

doi:10.1239/jap/1127322019

Cited by 55 publications

(88 citation statements)

References 30 publications

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“…where we have used independence of X i and X t , t = i in (8). Now extend from smooth f to absolutely continuous f where expectations in (2) exist by standard limiting arguments.…”

Section: Zero Biasing In One Dimensionmentioning

confidence: 99%

“…[15], [5], [13] and references therein). In [9] the authors introduced and studied the zero bias transformation in one dimension; further development is continued in [7] and [8]. Here the authors extend the study of zero biasing to R p , and illustrate with an application.…”

Section: Introductionmentioning

confidence: 99%

“…Due to the similarity between zero and size biasing, the transformation in Definition 1.1 was so coined as it offered a parallel of size biasing for mean zero variables, hence, zero biasing. For use of size biasing for normal approximation, see [11] and [8]; for families of distributional transformations of which both zero and size biasing are special cases, see [10]. In [9], after introducing the zero bias transformation, the authors apply it to show that smooth function rates of n −1 obtain for simple random sampling under certain higher order moment conditions.…”

Section: Introductionmentioning

confidence: 99%

“…In [9], after introducing the zero bias transformation, the authors apply it to show that smooth function rates of n −1 obtain for simple random sampling under certain higher order moment conditions. In [7] zero biasing is used to provide bounds to the normal for hierarchical sequences, and in [8] for normal approximation in combinatorial central limit theorems with random permutations having distribution constant over cycle type.…”

Section: Introductionmentioning

confidence: 99%

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Zero biasing in one and higher dimensions, and applications

Goldstein¹,

Reinert²

2005

Stein's Method and Applications

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Given any mean zero, finite variance σ 2 random variable W , there exists a unique distribution on a variable W * such that EW f (W ) = σ 2 Ef (W * ) for all absolutely continuous functions f for which these expectations exist. This distributional 'zero bias' transformation of W to W * , of which the normal is the unique fixed point, was introduced in [9] to obtain bounds in normal approximations. After providing some background on the zero bias transformation in one dimension, we extend its definition to higher dimension and illustrate with an application to the normal approximation of sums of vectors obtained by simple random sampling.

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“…where we have used independence of X i and X t , t = i in (8). Now extend from smooth f to absolutely continuous f where expectations in (2) exist by standard limiting arguments.…”

Section: Zero Biasing In One Dimensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Zero biasing in one and higher dimensions, and applications

Goldstein¹,

Reinert²

2005

Stein's Method and Applications

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“…Size biasing the number of local extrema on graphs, for the purpose of normal approximation, was studied in [1] and [12]. For a given graph G = {V, E}, let G v = {V v , E v }, v ∈ V, be a collection of isomorphic subgraphs of G such that v ∈ V v and for all v 1 , v 2 ∈ V the isomorphism from G v1 to G v2 maps v 1 to v 2 .…”

Section: Local Extrema On a Latticementioning

confidence: 99%

Concentration of measures via size-biased couplings

Ghosh

Goldstein

2009

Probab. Theory Relat. Fields

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Let Y be a nonnegative random variable with mean µ and finite positive variance σ 2 , and let Y s , defined on the same space as Y , have the Y size biased distribution, that is, the distribution characterized byfor all functions f for which these expectations exist.Under a variety of conditions on the coupling of Y and Y s , including combinations of boundedness and monotonicity, concentration of measure inequalities such asfor all t ≥ 0 hold for some explicit A and B. Examples include the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number of m-runs in a sequence of coin tosses, the number of local maximum of a random function on a lattice, the number of urns containing exactly one ball in an urn allocation model, the volume covered by the union of n balls placed uniformly over a volume n subset of R d , the number of bulbs switched on at the terminal time in the so called lightbulb process, and the infinitely divisible and compound Poisson distributions that satisfy a bounded moment generating function condition.

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Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one‐dimensional labels

Borga

2021

Random Struct Algorithms

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We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior of the consecutive patterns in random permutations studying properties of the consecutive increments in the corresponding random walks. The method applies to families of permutations with a one‐dimensional‐labeled generating tree (together with some technical assumptions) and implies local convergence for random permutations in such families. We exhibit ten different families of permutations, most of them being permutation classes, that satisfy our assumptions. To the best of our knowledge, this is the first work where generating trees—which were introduced to enumerate combinatorial objects—have been used to establish probabilistic results.

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Berry-Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing

Cited by 55 publications

References 30 publications

Zero biasing in one and higher dimensions, and applications

Zero biasing in one and higher dimensions, and applications

Concentration of measures via size-biased couplings

Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one‐dimensional labels

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