Berry Esseen Bounds for Combinatorial Central Limit Theorems and Pattern Occurrences, using Zero and Size Biasing

By Larry Goldstein. 2005

76 citations recorded by the source corpus.

Berry-Esseen-type bounds to the normal, based on zero- and size-bias couplings, are derived using Stein's method. The zero biasing bounds are illustrated in an application to combinatorial central limit theorems in which the random permutation has either the uniform distribution or one that is constant over permutations with the same cycle type, with no fixed points. The size biasing bounds are applied to the occurrences of fixed, relatively ordered subsequences (such as rising sequences) in a random permutation, and to the occurrences of patterns, extreme values, and subgraphs in finite graphs.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/ac81bc8f990cf803644382beef89bf449c58b417

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In Artificial Intelligence

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

Bounds on the constant in the mean central limit theorem (2009, background): “How closeness to normality may be measured by the closeness of a distribution to its zero bias transform, and related applications, are the topics of [5, 6] and [7].”
New rates for exponential approximation and the theorems of Rényi and Yaglom (2011, methodology): “tion. It was also (independently of the current work) previously used by Bon (2006, Lemma 6) to study geometric convolutions. Whereas zero-bias couplings are diﬃcult to construct in general (see e.g. Goldstein (2005), Goldstein (2007) and Ghosh (2009)), it turns out that there is a convenient way to”
Stein couplings for normal approximation (2010, background): “In contrast, in the zero-biasing approach as introduced by Goldstein and Reinert (1997), it is assumed that (1.”
Stein's method for normal approximation (2005, methodology): “We also refer to Goldstein & Reinert (1997) for some of the important features of the zero bias transformation and to Goldstein (2005) for Berry–Esseen bounds for combinatorial central limit theorems and pattern occurrences using zero and size biasing approaches.”
Fundamentals of Stein's method (2011, methodology): “36 can be applied to sums of independent random variables which are almost surely bounded (note that W bounded implies W z bounded), and can also be used to derive a bound in Hoeffding’s combinatorial CLT under some boundedness assumption [30].”
Distributional Transformations, Orthogonal Polynomials, and Stein Characterizations (2005, unknown): “ar X iv :m at h/ 05 10 24 0v 1 [ m at h. PR ] 1 2 O ct 2”
From Stein identities to moderate deviations (2009, unknown): “For normal approximation there are four methods for constructing a Stein identity: the direct method (Stein (1972)), zero-bias coupling (Goldstein and Reinert (1997) and Goldstein (2005)), exchangeable pairs (Stein (1986)), and Stein coupling (Chen and Röllin (2010)).”