, Volume 21, Issue 2, pp 335-352

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

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Abstract

Let Λ = {λ k } be an infinite increasing sequence of positive integers with λ k →∞. Let X = {X(t), t ∈¸ R N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K 1 and K 2 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) = s N/α (log log 1/s) N/(2α), ϕ-p Λ(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.

Supported by the National Natural Science Foundation of China (No.10471148), Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei (No.2003A005)