Billingsley’s packing dimension

Abstract

For a stochastic process on a finite state space, we define the notion of a packing measure based on the naturally defined cylinder sets. For any two measures ν \nu , γ \gamma , corresponding to the same stochastic process, if \[ F ⊆ { ω ∈ Ω : lim n log ⁡ γ ( c n ( ω ) ) log ⁡ ν ( c n ( ω ) ) = θ } , F \subseteq \left \{ \omega \in \Omega : \lim _{n} \frac {\log \gamma (c_{n}(\omega ))}{\log \nu (c_{n}(\omega ))} = \theta \right \}, \] then we prove that \[ D i m ν ( F ) = θ   D i m γ ( F ) . {\rm {Dim}}_{\nu }(F) = \theta ~{\rm {Dim}}_{\gamma }(F). \]