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

Abstract

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 with an application to combinatorial central limit theorems where the random permutation has either the uniform distribution or one which is constant over permutations with the same cycle type and having no fixed points. The size biasing bounds are applied to the occurrences of fixed relatively ordered sub-sequences (such as rising sequences) in a random permutation, and to the occurrences of patterns, extreme values, and subgraphs on finite graphs.

23 pages

References in corpus (4)

• Normal approximation under local dependence
• Stein's method and the zero bias transformation with application to simple random sampling
• Normal approximation for hierarchical structures
• A permutation test for matching and its asymptotic distribution

Cited by in corpus (27)

• Stein couplings for normal approximation
• From Stein identities to moderate deviations
• Bounds on the constant in the mean central limit theorem
• $L^1$ bounds in normal approximation
• On the error bound in a combinatorial central limit theorem
• Normal approximation for coverage models over binomial point processes
• Esseen type bounds of the remainder in a combinatorial CLT
• Zero biasing and a discrete central limit theorem
• Concentration inequalities via zero bias couplings
• A Berry-Esseen bound with applications to vertex degree counts in the Erdős-Rényi random graph
• A central limit theorem for descents of a Mallows permutation and its inverse
• Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one-dimensional labels
• Functional approximations with Stein's method of exchangeable pairs
• A Berry Esseen Theorem for the Lightbulb Process
• Asymptotic Normality of Random Sums of m-dependent Random Variables
• A Berry-Esseen bound for the uniform multinomial occupancy model
• Shuffling cards for blackjack, bridge, and other card games
• Concentration of measures via size biased couplings
• Zero biasing and growth processes
• Error bounds in local limit theorems using Stein's method
• Distributional transformations, orthogonal polynomials, and Stein characterizations
• Normal approximations for descents and inversions of permutations of multisets
• Bounds of the remainder in a combinatorial central limit theorem
• Multivariate concentration of measure type results using exchangeable pairs and size biasing
• Moments of permutation statistics and central limit theorems
• $L^p$ bounds for a central limit theorem with involutions
• Normal approximation for isolated balls in an urn allocation model