L ^∞: Applications

Abstract

Chapter 6 applies the L ^∞ results of Chap. 5 to a number of applications, all of which involve dependence. The chapter deals mainly with global dependence but begins to also touch upon local dependence, a topic more thoroughly explored in Chap. 9. Regarding global dependence, the analysis of the combinatorial central limit theorem, studied in L ¹ in Chap. 4, is continued here with the goal of obtaining L ^∞ results. Results for the classical case are given, where the permutation is uniformly chosen over the symmetric group, as well as for the case where the permutation is chosen with distribution constant over some conjugacy class, such as the class of involutions. Two approaches are considered, one using the zero bias coupling and one using induction. Normal approximation bounds for the so called lightbulb process are also given in this chapter, again an example of handling global dependence, this time by the size bias coupling. The anti-voter model is studied using the exchangeable pair technique, as is the binary expansion of a random integer. Results for the occurrences of patterns in graphs and permutations, an example of local dependence, are considered using the size bias method.