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On the convergence of generalized moments in almost sure central limit theorem

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  • Ibragimov, Ildar
  • Lifshits, Mikhail

Abstract

Let {[zeta]k} be the normalized sums corresponding to a sequence of i.i.d. variables with zero mean and unit variance. Define random measures and let G be the normal distribution. We show that for each continuous function h satisfying [integral operator] hdG

Suggested Citation

  • Ibragimov, Ildar & Lifshits, Mikhail, 1998. "On the convergence of generalized moments in almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 343-351, November.
    • Handle: RePEc:eee:stapro:v:40:y:1998:i:4:p:343-351
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    References listed on IDEAS

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    1. 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.
    2. Berkes, István & Csáki, Endre & Horváth, Lajos, 1998. "Almost sure central limit theorems under minimal conditions," Statistics & Probability Letters, Elsevier, vol. 37(1), pages 67-76, January.
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    Cited by:

    1. Blacher, René, 2007. "Central Limit Theorem by moments," Statistics & Probability Letters, Elsevier, vol. 77(17), pages 1647-1651, November.
    2. Berkes, István & Horváth, Lajos, 2001. "The logarithmic average of sample extremes is asymptotically normal," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 77-98, January.
    3. Berkes, István & Csáki, Endre, 2001. "A universal result in almost sure central limit theory," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 105-134, July.
    4. Bercu, Bernard & Nourdin, Ivan & Taqqu, Murad S., 2010. "Almost sure central limit theorems on the Wiener space," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1607-1628, August.
    5. Wu, Qunying, 2011. "Almost sure limit theorems for stable distributions," Statistics & Probability Letters, Elsevier, vol. 81(6), pages 662-672, June.
    6. Fahrner, Ingo, 2000. "An extension of the almost sure max-limit theorem," Statistics & Probability Letters, Elsevier, vol. 49(1), pages 93-103, August.
    7. Bercu, B., 2004. "On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 157-173, May.
    8. Zhicheng Chen & Hongyun Zhang & Xinsheng Liu, 2020. "Almost Sure Convergence for the Maximum and Minimum of Normal Vector Sequences," Mathematics, MDPI, vol. 8(4), pages 1-11, April.
    9. 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.
    10. Xu, Feng & Wu, Qunying, 2017. "Almost sure central limit theorem for self-normalized partial sums of ρ−-mixing sequences," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 17-27.
    11. Berkes, István, 2001. "The law of large numbers with exceptional sets," Statistics & Probability Letters, Elsevier, vol. 55(4), pages 431-438, December.

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