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On strong versions of the central limit theorem

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  • Rychlik, Zdzislaw
  • Szuster, Konrad S.

Abstract

The purpose of this paper is to present strong versions of the central limit theorem for independent nonidentically distributed random variables. Let Sn, n[greater-or-equal, slanted]1, be the partial sums of independent random variables with zero means and finite variances and let a(x) be a real function. We present sufficient conditions under which the arithmetic means of a(Snk/(ESnk2)1/2) converge almost surely to for some subsequences {nk, k[greater-or-equal, slanted]1}, where [Phi] denotes the standard normal distribution.

Suggested Citation

  • Rychlik, Zdzislaw & Szuster, Konrad S., 2003. "On strong versions of the central limit theorem," Statistics & Probability Letters, Elsevier, vol. 61(4), pages 347-357, February.
    • Handle: RePEc:eee:stapro:v:61:y:2003:i:4:p:347-357
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    References listed on IDEAS

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    1. Peligrad, Magda & Shao, Qi-Man, 1995. "A note on the almost sure central limit theorem for weakly dependent random variables," Statistics & Probability Letters, Elsevier, vol. 22(2), pages 131-136, February.
    2. Lacey, Michael T. & Philipp, Walter, 1990. "A note on the almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 201-205, March.
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