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

Let Λ={λ} be an infinite increasing sequence of positive integers with λ→∞. 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<ad, then there exist positive finite constants Kand Ksuch that, with unit probability, if and only if there exists γ>0 such that where φ(s)=s(loglog 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));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.