Strong laws for associated random variables

ABSTRACT

We study the almost sure convergence and rates of weighted sums of associated random variables under the classical assumption of existence of Laplace transforms. This assumption implies the existence of every moment, so we address the same problem assuming a suitable decrease rate on tail joint probabilities which only implies the existence of finitely many moments, proving the analogous characterizations of convergence and rates. Still relaxing further the assumptions on moment existence, we also prove a Marcinkiewicz–Zygmund for associated variables without means, complementing existing results for this dependence structure.

KEYWORDS:

• Almost sure convergence
• association
• convergence rates
• exponential inequalities
• Marcinkiewicz–Zygmund law

AMS SUBJECT CLASSIFICATION:

• 60F15

Acknowledgments

The authors wish to thank the anonymous referee and the Associate Editor whose careful reading and suggestions helped improving on an earlier versions of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

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 – Fundação para a Ciência e a Tecnologia/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020 [PEst-C/MAT/UI0324/2013]