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Laws of the iterated logarithm for weighted sums of independent random variables


  • Li, Deli
  • Tomkins, R. J.


Let [Lambda] = lim supn-->[infinity](2n log log n)-1/2 [Sigma]k=1n[latin small letter f with hook](k/n)Xk, where [latin small letter f with hook] is a function defined on [0,1] and {X, Xn;n[greater-or-equal, slanted]1} is an iid sequence. If X is real-valued, it is shown that [Lambda] = [latin small letter f with hook]2, the L2-norm of [latin small letter f with hook], for all functions [latin small letter f with hook] in a certain class of absolutely continuous functions if E(X) = 0 and E(X2) = 1. Conversely, if [Lambda] = [latin small letter f with hook]2 for some such [latin small letter f with hook] with [integral operator]01[latin small letter f with hook](t)dt [not equal to] 0, then E(X) = 0, E(X2) = 1. Necessary and sufficient conditions for the compact law of the iterated logarithm are given in the case when X takes values in a separable Banach space, and a law of the iterated logarithm for sums of weighted partial sums is obtained in a Banach space setting.

Suggested Citation

  • Li, Deli & Tomkins, R. J., 1996. "Laws of the iterated logarithm for weighted sums of independent random variables," Statistics & Probability Letters, Elsevier, vol. 27(3), pages 247-254, April.
  • Handle: RePEc:eee:stapro:v:27:y:1996:i:3:p:247-254

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    Cited by:

    1. Peng, Liang & Qi, Yongcheng, 2003. "Chover-type laws of the iterated logarithm for weighted sums," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 401-410, December.
    2. Li, Deli & Qi, Yongcheng & Rosalsky, Andrew, 2009. "Iterated logarithm type behavior for weighted sums of i.i.d. random variables," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 643-651, March.
    3. Pingyan, Chen, 2002. "Limiting behavior of weighted sums with stable distributions," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 367-375, December.
    4. Li, Deli & Bhaskara Rao, M. & Tomkins, R. J., 2001. "The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics," Journal of Multivariate Analysis, Elsevier, vol. 78(2), pages 191-217, August.
    5. Zuoxiang Peng & Zhongquan Tan & Saralees Nadarajah, 2011. "Almost sure central limit theorem for the products of U-statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 73(1), pages 61-76, January.


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