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On the asymptotic behavior of randomly weighted averages

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  • Roozegar, Rasool
  • Soltani, A.R.

Abstract

In this paper we assume that X1,X2,… is a sequence of independent continuous centered random variables with finite variances σ12,σ22,…. Then we present a central limit theorem for the randomly weighted averages Sn=R1X1+⋯+RnXn, where the random weights R1,…,Rn are the cuts of (0,1) by the order statistics of a random sample of size n−1 from a uniform distribution on (0,1). Indeed we prove that under certain assumptions on the variances, n+1Sn converges in distribution to the normal distribution with mean zero and variance 2c, c=limn→∞(1/n)∑i=1nσi2.

Suggested Citation

  • Roozegar, Rasool & Soltani, A.R., 2015. "On the asymptotic behavior of randomly weighted averages," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 269-272.
    • Handle: RePEc:eee:stapro:v:96:y:2015:i:c:p:269-272
    DOI: 10.1016/j.spl.2014.10.003
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    References listed on IDEAS

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    1. Yang, Yang & Leipus, Remigijus & Šiaulys, Jonas, 2012. "Tail probability of randomly weighted sums of subexponential random variables under a dependence structure," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1727-1736.
    2. Rosalsky, Andrew & Sreehari, M., 1998. "On the limiting behavior of randomly weighted partial sums," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 403-410, November.
    3. Liu, Quansheng, 2001. "Asymptotic properties and absolute continuity of laws stable by random weighted mean," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 83-107, September.
    4. 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.
    5. Weber, Michel, 2006. "A weighted central limit theorem," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1482-1487, August.
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    Cited by:

    1. 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).

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