Almost Sure Convergence

TLDR: In this paper, the authors study the Strong Laws of Large Numbers (SLLN) for associated variables and their applications to the characterization of asymptotics of statistical estimators under associated sampling.

Abstract: This chapter studies essentially Strong Laws of Large Numbers (SLLN) for associated variables and their applications to the characterization of asymptotics of statistical estimators under associated sampling. It is possible to prove SLLN under fairly general assumptions, but, in order to prove characterizations of convergence rates, a closer care on the control of the covariances, based on the inequalities studied in the previous chapter, is required. Sect. 3.2 handles this kind of results, proving almost optimal convergence rates, that is, convergence rates arbitrarily close to those for independent variables. There exist characterizations of convergence rates based on extensions of the Law of Iterated Logarithm to associated variables. Such results are deferred to Chap. 4, as their proofs require a few inequalities to be proved there. We include a section on large deviations, a not yet very explored theme under association. Here the assumptions on the decay rate of the covariances are much stronger, a behaviour as found for some other dependence structures. The approach and techniques used in this chapter are adapted in the final section to prove almost sure consistency results for nonparametric density and regression estimators based on associated samples.