Packing-type measures of the sample paths of fractional brownian motion

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, K1 ≤ φ - p∧ (X([0,1])N) ≤ φ - p∧(GrX([0,1]N)) ≤ K2 if and only if there exixts γ > 0 such that Σk=1∞ 1/λkγ = ∞, where φ (s) = s N/α(log log 1/s)N/(2α), φ-p∧ (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. © Springer-Verlag 2005.