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Bounded and compact laws of the logarithm for B-valued random variables

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  • Li, Deli

Abstract

In this paper, we study a version of the law of the logarithm in a Banach space setting. Some necessary and some sufficient conditions are presented for the law of the logarithm for B-valued random variables. The law of the logarithm, the law of the iterated logarithm and the central limit theorem are shown to be equivalent for finite-dimentional B-valued random variables. However, this statement is not true for infinite-dimensional case. Under the central limit theorem, the law of the logarithm is shown to be equivalent to some certain moment condition. The law of the iterated logarithm implies the law of the logarithm, but the converse is not true. All methods used in this paper are quite standard in probability in Banach spaces except for some modifications. We made an effort to solve this problem completely in a Banach space using both the isoperimetric methods and the Gaussian randomization technique, but we were not successful.

Suggested Citation

  • Li, Deli, 1996. "Bounded and compact laws of the logarithm for B-valued random variables," Stochastic Processes and their Applications, Elsevier, vol. 63(2), pages 189-209, November.
    • Handle: RePEc:eee:spapps:v:63:y:1996:i:2:p:189-209
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    References listed on IDEAS

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    1. Li, D. L. & Rao, M. B. & Wang, X. C., 1995. "On the Strong Law of Large Numbers and the Law of the Logarithm for Weighted Sums of Independent Random Variables with Multidimensional Indices," Journal of Multivariate Analysis, Elsevier, vol. 52(2), pages 181-198, February.
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    Cited by:

    1. Chen, Pingyan & Chen, Ran, 2010. "A remark on LSL for weighted sums of i.i.d random elements," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1329-1334, September.
    2. D. Li & R. J. Tomkins, 2003. "The Law of the Logarithm for Weighted Sums of Independent Random Variables," Journal of Theoretical Probability, Springer, vol. 16(3), pages 519-542, July.
    3. Sung, Soo Hak, 2009. "A law of the single logarithm for weighted sums of i.i.d. random elements," Statistics & Probability Letters, Elsevier, vol. 79(10), pages 1351-1357, May.

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    Keywords

    60B12 60F15 G0G50;

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