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On the limiting behavior of randomly weighted partial sums

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  • Rosalsky, Andrew
  • Sreehari, M.

Abstract

We study the almost sure limiting behavior and convergence in probability of weighted partial sums of the form where {Wnj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n, n[less-than-or-equals, slant]1} and {Xnj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n, n[greater-or-equal, slanted]1} are triangular arrays of random variables. The results obtain irrespective of the joint distributions of the random variables within each array. Applications concerning the Efron bootstrap and queueing theory are discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:40:y:1998:i:4:p:403-410
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    References listed on IDEAS

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    1. Arenal-Gutiérrez, Eusebio & Matrán, Carlos & Cuesta-Albertos, Juan A., 1996. "On the unconditional strong law of large numbers for the bootstrap mean," Statistics & Probability Letters, Elsevier, vol. 27(1), pages 49-60, March.
    2. Cabrera, M. Ordóñez, 1988. "Limit theorems for randomly weighted sums of random elements in normed linear spaces," Journal of Multivariate Analysis, Elsevier, vol. 25(1), pages 139-145, April.
    3. Rosalsky, Andrew, 1993. "On the almost certain limiting behavior of normed sums of identically distributed positive random variables," Statistics & Probability Letters, Elsevier, vol. 16(1), pages 65-70, January.
    4. Wei, Duan & Taylor, R. L., 1978. "Convergence of weighted sums of tight random elements," Journal of Multivariate Analysis, Elsevier, vol. 8(2), pages 282-294, June.
    5. Adler, André & Rosalsky, Andrew & Taylor, Robert L., 1991. "A weak law for normed weighted sums of random elements in rademacher type p banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 37(2), pages 259-268, May.
    6. Arenal-Gutiérrez, Eusebio & Matrán, Carlos & Cuesta-Albertos, Juan A., 1996. "Unconditional Glivenko-Cantelli-type theorems and weak laws of large numbers for bootstrap," Statistics & Probability Letters, Elsevier, vol. 26(4), pages 365-375, March.
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    Cited by:

    1. Pingyan, Chen & Shixin, Gan, 2007. "Limiting behavior of weighted sums of i.i.d. random variables," Statistics & Probability Letters, Elsevier, vol. 77(16), pages 1589-1599, October.
    2. Shen, Xinmei & Lin, Zhengyan, 2008. "Precise large deviations for randomly weighted sums of negatively dependent random variables with consistently varying tails," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3222-3229, December.
    3. Hu, Tien-Chung & Cabrera, Manuel Ordóñez & Volodin, Andrei I., 2001. "Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 155-164, January.
    4. A. Elijio & V. Čekanavičius, 2015. "Compound Poisson approximation to weighted sums of symmetric discrete variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(1), pages 195-210, February.
    5. Sung, Soo Hak, 2001. "Strong laws for weighted sums of i.i.d. random variables," Statistics & Probability Letters, Elsevier, vol. 52(4), pages 413-419, May.
    6. Roozegar, Rasool & Soltani, A.R., 2015. "On the asymptotic behavior of randomly weighted averages," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 269-272.
    7. Csörgő, Miklós & Nasari, Masoud M., 2013. "Asymptotics of randomly weighted u- and v-statistics: Application to bootstrap," Journal of Multivariate Analysis, Elsevier, vol. 121(C), pages 176-192.
    8. Cabrera, Manuel Ordóñez & Volodin, Andrei I., 2001. "On conditional compactly uniform pth-order integrability of random elements in Banach spaces," Statistics & Probability Letters, Elsevier, vol. 55(3), pages 301-309, December.

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