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

DOI:
https://doi.org/10.1090/tpms/1077

Published electronically:
February 27, 2020

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Abstract | References | Similar Articles | Additional Information

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

Email:
idir@mat.uc.pt

**P. E. Oliveira**

Affiliation:
CMUC, Department of Mathematics, University of Coimbra, Portugal

Email:
paulo@mat.uc.pt

DOI:
https://doi.org/10.1090/tpms/1077

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