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Local limit theorems for occupancy models
Author(s) -
Barbour A. D.,
Braunsteins Peter,
Ross Nathan
Publication year - 2021
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20967
Subject(s) - mathematics , limit (mathematics) , central limit theorem , stein's method , logarithm , rate of convergence , convergence (economics) , random graph , graph , combinatorics , discrete mathematics , statistics , mathematical analysis , computer science , metric space , intrinsic metric , channel (broadcasting) , computer network , convex metric space , economics , economic growth
We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with d neighbors in a germ‐grain model, and the number of degree‐ d vertices in an Erdős‐Rényi random graph. In both cases, the error rate is optimal, up to logarithmic factors.