Local limit theorems for occupancy models. Issue 1 (27th September 2020)

• Record Type: Journal Article
• Authors: Barbour, A. D.; Braunsteins, Peter; Ross, Nathan
• Is Part Of: Random structures & algorithms. Volume 58:Issue 1(2021)
• Page Start: 3
• Page End: 33
• Publication Date: 2020-09-27
• Subjects: Erdös‐Renyi random graph -- germ‐grain models -- local limit theorems -- Stein's method; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20967
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : 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.