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Precise rates in the law of the iterated logarithm for R/S statistics

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Abstract

Let {X n; n ≥ 1} be a sequence of i.i.d. random variables with finite variance, Q(n) be the related R/S statistics. It is proved that

$$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^2 \sum\limits_{n = 1}^\infty {\frac{1}{{n log n}}P\left\{ {Q(n) \geqslant \varepsilon \sqrt {2n log log n} } \right\} = \frac{1}{2}EY^2 } $$

, where Y = sup0<t≤1 B(t) − inf0≤t≤s B(t), and B(t) is a Brownian bridge.

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Authors and Affiliations

  1. City College of Zhejiang University, Hangzhou, 310015, China

    Wu Hongmei

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Wen Jiwei

Authors
  1. Wu Hongmei
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  2. Wen Jiwei
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Supported by the NNSF of China(10071072).

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Wu, H., Wen, J. Precise rates in the law of the iterated logarithm for R/S statistics. Appl. Math. Chin. Univ. 21, 461–466 (2006). https://doi.org/10.1007/s11766-006-0010-7

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  • DOI: https://doi.org/10.1007/s11766-006-0010-7

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