A sequence of elements an of R d converges to a limit a if and only if, for each ǫ > 0, the sequence an eventually lies within a ball of radius ǫ centered at a. It’s okay if the first few (or few million) terms lie outside that ball— and the number of terms that do lie outside the ball may depend on how big ǫ is (if ǫ is small enough it typically will take millions of terms before the remaining...

Let X(t) (t ∈ R) be a fractional Brownian motion of index α in Rd. If 1 < αd , then there exists a positive finite constant K such that with probability 1, φ-p(X([0, t])) = Kt for any t > 0 , where φ(s) = s 1 α /(log log 1 s ) 1 2α and φ-p(X([0, t])) is the φ-packing measure of X([0, t]).