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Theory of Probability and Mathematical Statistics

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Asymptotic results for certain weak dependent variables

Authors: I. Arab and P. E. Oliveira
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 99 (2018).
Journal: Theor. Probability and Math. Statist. 99 (2019), 19-37
MSC (2010): Primary 60F10; Secondary 60F05, 60F17
Published electronically: February 27, 2020
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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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Additional Information

I. Arab
Affiliation: CMUC, Department of Mathematics, University of Coimbra, Portugal

P. E. Oliveira
Affiliation: CMUC, Department of Mathematics, University of Coimbra, Portugal

Keywords: Central Limit Theorem, convergence rate, L-weak dependence, Strong Law of Large Numbers.
Received by editor(s): June 27, 2018
Published electronically: February 27, 2020
Additional Notes: This work was partially supported by the Centre for Mathematics of the University of Coimbra — UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.
Article copyright: © Copyright 2020 American Mathematical Society