Asymptotic results for certain weak dependent variables

Arab, I.; Oliveira, P.

doi:10.1090/tpms/1077

Asymptotic results for certain weak dependent variables

Authors: I. Arab and P. E. Oliveira
Journal: Theor. Probability and Math. Statist. 99 (2019), 19-37
MSC (2010): Primary 60F10; Secondary 60F05, 60F17.
DOI: https://doi.org/10.1090/tpms/1077
Published electronically: February 27, 2020
MathSciNet review: 3908653
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Abstract: We consider a special class of weak dependent random variables with control on covariances of Lipschitz transformations. This class includes, but is not limited to, positively, negatively associated variables and a few other classes of weakly dependent structures. We prove the Strong Law of Large Numbers with the characterization of convergence rates which is almost optimal, in the sense that it is arbitrarily close to the optimal rate for independent variables. Moreover, we prove an inequality comparing the joint distributions with the product distributions of the margins, similar to the well-known Newman’s inequality for characteristic functions of associated variables. As a consequence, we prove the Central Limit Theorem together with its functional counterpart, and also the convergence of the empirical process for this class of weak dependent variables.

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