L₁-Convergence for Weighted Sums of Some Dependent Random Variables

Abstract

In this article, the authors discuss the L ₁-convergence for weighted sums of some dependent random variables under the condition of h-integrability with respect to an array of weights. The dependence structure of the random variables includes pairwise lower case negative dependence and conditions on the mixing coefficient, the maximal correlation coefficient, or the ρ*-mixing coefficient. They prove that all the weighted sums have similar limiting behaviour.

Keywords:

• Cesàro uniform integrability
• h-integrability
• L ₁-convergence
• maximal correlation coefficient
• Mixing coefficient
• Pairwise lower case negatively dependent random variables
• ρ*-mixing coefficient

Mathematics Subject Classification:

• 60F05
• 60F15

The authors are grateful to the referee for carefully reading the manuscript and for providing some comments and suggestions, which led to improvements in this article.

The research of P.C. has been supported by the National Natural Science Foundation of China.

The research of M.O.C. has been partially supported by DGICYT grant BFM2003-03893-C02-01 and Junta de Andalucia FQM 127.

The research of A.V. has been partially supported by the National Sciences and Engineering Research Council of Canada.