Typicality for stratified measures

Abstract

Stratified measures on Euclidean space are defined here as convex combinations of rectifiable measures. They are possibly singular with respect to the Lebesgue measure and generalize continuous-discrete mixtures. A stratified measure $ρ$ can thus be represented as $\sum_{i=1}^k q_i ρ_i$, where $(q_1,..,q_k)$ is a probability vector and each $ρ_i$ is $m_i$-rectifiable for some integer $m_i$ i.e. absolutely continuous with respect to the $m_i$-Hausdorff measure $μ_i$ on a $m_i$-rectifiable set $E_i$ (e.g. a smooth $m_i$-manifold). We introduce a set of strongly typical realizations of $ρ^{\otimes n}$ (memoryless source) that occur with high probability. The typical realizations are supported on a finite union of strata $\{E_{i_1}\times \cdots \times E_{i_n}\}$ whose dimension concentrates around the mean dimension $\sum_{i=1}^k q_i m_i$. For each $n$, an appropriate sum of Hausdorff measures on the different strata gives a natural notion of reference "volume"; the exponential growth rate of the typical set's volume is quantified by Csiszar's generalized entropy of $ρ$ with respect to $μ=\sum_{i=1}^k μ_i$. Moreover, we prove that this generalized entropy satisfies a chain rule and that the conditional term is related to the volume growth of the typical realizations in each stratum. The chain rule and its asymptotic interpretation hold in the more general framework of piecewise continuous measures: convex combinations of measures restricted to pairwise disjoint sets equipped with reference $σ$-finite measures. Finally, we establish that our notion of mean dimension coincides with Rényi's information dimension when applied to stratified measures, but the generalized entropy used here differs from Rényi's dimensional entropy.