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

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.