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On the asymptotic distribution of randomly weighted averages of random vectors

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  • Roozegar, Rasool
  • zarch, Hamid Reza Taherizadeh

Abstract

Let X1,X2,…,Xn be a sequence of mutually independent continuous random vectors with respective positive definite covariance matrices Σ1,Σ2,…,Σn. The main purpose of this article is to derive the asymptotic distribution of Sn:n, a randomly weighted average of the sequence X1,X2,…,Xn, as n→∞. The random weights are the cuts of (0, 1) by an increasing ordered array of the ordered statistics of n independent and identically uniformly distributed random variables. We prove that under certain assumptions on the covariance matrices, n(Sn:n−μ) converges in distribution to the multivariate normal distribution with zero mean and covariance matrix 2Σ, where Σ is the limit of the expression in terms of Σis. Finally, we give an application for the multivariate randomly weighted averages in modeling (height, general intelligence) parents’ gene compositions given to their offspring. The simulation results which are based on three distributions (Bivariate t, Dirichlet and Normal) which confirm the main theorem of the paper.

Suggested Citation

  • Roozegar, Rasool & zarch, Hamid Reza Taherizadeh, 2021. "On the asymptotic distribution of randomly weighted averages of random vectors," Statistics & Probability Letters, Elsevier, vol. 179(C).
    • Handle: RePEc:eee:stapro:v:179:y:2021:i:c:s0167715221001930
    DOI: 10.1016/j.spl.2021.109231
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    References listed on IDEAS

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    1. Roozegar, Rasool & Soltani, A.R., 2015. "On the asymptotic behavior of randomly weighted averages," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 269-272.
    2. Soltani, Ahmad Reza & Roozegar, Rasool, 2012. "On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 1012-1020.
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    Cited by:

    1. Thomas Hitchen & Saralees Nadarajah, 2024. "Exact Results for the Distribution of Randomly Weighted Sums," Mathematics, MDPI, vol. 12(1), pages 1-22, January.

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