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Some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables

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Abstract

In (Ordóñez Cabrera and Volodin, J. Math. Anal. Appl. 305:644–658, 2005), the authors introduce the notion of h-integrability of an array of random variables with respect to an array of constants, and obtained some mean convergence theorems for weighted sums of random variables subject to some special kinds of dependence.

In view of the important role played by conditioning and dependence in the models used to describe many situations in the applied sciences, the concepts and results in the aforementioned paper are extended herein to the case of randomly weighted sums of dependent random variables when a sequence of conditioning sigma-algebras is given. The dependence conditions imposed on the random variables (conditional negative quadrant dependence and conditional strong mixing) as well as the convergence results obtained are conditional relative to the conditioning sequence of sigma-algebras.

In the last section, a strong conditional convergence theorem is also established by using a strong notion of conditional h-integrability.

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Authors and Affiliations

  1. Department of Mathematical Analysis, University of Sevilla, Sevilla, 41080, Spain

    Manuel Ordóñez Cabrera

  2. Department of Statistics, University of Florida, Gainesville, FL, 32611, USA

    Andrew Rosalsky

  3. Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S0A2, Canada

    Andrei Volodin

Authors
  1. Manuel Ordóñez Cabrera
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  2. Andrew Rosalsky
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  3. Andrei Volodin
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Corresponding author

Correspondence to Andrew Rosalsky.

Additional information

The research of M. Ordóñez Cabrera has been partially supported by DGICYT grant BFM2003-03893-C02-01 and Junta de Andalucia FQM 127. The research of A. Volodin has been partially supported by the National Science and Engineering Research Council of Canada.

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Ordóñez Cabrera, M., Rosalsky, A. & Volodin, A. Some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables. TEST 21, 369–385 (2012). https://doi.org/10.1007/s11749-011-0248-0

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  • DOI: https://doi.org/10.1007/s11749-011-0248-0

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