Acta Mathematicae Applicatae Sinica

, Volume 21, Issue 2, pp 335–352

Packing-type Measures of the Sample Paths of Fractional Brownian Motion

Original Papers

DOI: 10.1007/s10255-005-0241-z

Cite this article as:
Chen, Z., Liu, S. & Xu, C. Acta Mathematicae Applicatae Sinica, English Series (2005) 21: 335. doi:10.1007/s10255-005-0241-z


Let Λ = {λk} be an infinite increasing sequence of positive integers with λk→∞. Let X = {X(t), t ∈¸ RN} be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in Rd. Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K1 and K2 such that, with unit probability,
$$ K_{1} \leqslant \varphi - p \wedge {\left( {X{\left( {{\left[ {0,1} \right]}} \right)}^{N} } \right)} \leqslant \varphi - p \wedge {\left( {GrX{\left( {{\left[ {0,1} \right]}^{N} } \right)}} \right)} \leqslant K_{2} $$
if and only if there exists γ > 0 such that
$$ {\sum\limits_{k = 1}^\infty {\frac{1} {{\lambda ^{\gamma }_{k} }}} } = \infty , $$
where ϕ(s) = sN/α(log log 1/s)N/(2α), ϕ-pΛ(E) is the Packing-type measure of E,X([0, 1])N is the image and GrX([0, 1]N) = {(t,X(t)); ¸ [0, 1]N} is the graph of X, respectively. We also establish liminf type laws of the iterated logarithm for the sojourn measure of X.


Fractional Brownian motion packing-type measure image graph law of iterated logarithm sojourn measure 

2000 MR Subject Classification

60G15 60G17 

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Applied MathematicsXidian UniversityXi’an 710071China
  2. 2.School of Information and MathematicsYangtze UniversityHubei 434104China
  3. 3.School of Mathematics and Computer ScienceCentral University for NationalitiesBeijing 100081China

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