随机性和维数的发散公式
A Divergence Formula for Randomness and Dimension
摘要：
如果$S$是有限字母表$Σ$上的无限序列，$β$是$Σ$上的概率测度，则$ S$关于$β$的{ \it维数}，写作$\dim^β(S)$，是比林斯利维数的构造版本，当$β$是一致概率测度时，它与（构造Hausdorff）维数$\dim(S)$一致。本文证明了$\dim^β(S)$及其对偶$\Dim^β(S)$，即$S$关于$β$的{ \it强维数}，可以与随机性一起用来度量$S$上的两个概率测度$α$和$β$的相似性.具体地说，我们证明了{ \it发散公式} \[ \dim^β(R) = \Dim^β(R) =\frac{\CH(α)}{\CH(α) + \D(α|| β)} \]持有$α$和$β$是可计算的，积极的概率措施$Σ$和$R \in Σ^\infty$是随机的$α$。在这个公式中，$\CH(α)$是$α$的香农熵，$\D(α||β)$是$α$和$β$之间的Kullback-Leibler散度。我们还表明，上述公式适用于所有序列$R$是$α$ -正常的（在Borel的意义上）时，$\dim^β(R)$和$\Dim^β(R)$取代更有效的有限状态尺寸$\dimfs^β(R)$和$\Dimfs^β(R)$。在证明这一点的过程中，我们还证明了有限状态压缩特性的$\dimfs^β(S)$和$\Dimfs^β(S)$。
英文摘要：
If $S$ is an infinite sequence over a finite alphabet $Σ$ and $β$ is a probability measure on $Σ$, then the {\it dimension} of $ S$ with respect to $β$, written $\dim^β(S)$, is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension $\dim(S)$ when $β$ is the uniform probability measure. This paper shows that $\dim^β(S)$ and its dual $\Dim^β(S)$, the {\it strong dimension} of $S$ with respect to $β$, can be used in conjunction with randomness to measure the similarity of two probability measures $α$ and $β$ on $Σ$. Specifically, we prove that the {\it divergence formula} \[ \dim^β(R) = \Dim^β(R) =\frac{\CH(α)}{\CH(α) + \D(α|| β)} \] holds whenever $α$ and $β$ are computable, positive probability measures on $Σ$ and $R \in Σ^\infty$ is random with respect to $α$. In this formula, $\CH(α)$ is the Shannon entropy of $α$, and $\D(α||β)$ is the Kullback-Leibler divergence between $α$ and $β$. We also show that the above formula holds for all sequences $R$ that are $α$-normal (in the sense of Borel) when $\dim^β(R)$ and $\Dim^β(R)$ are replaced by the more effective finite-state dimensions $\dimfs^β(R)$ and $\Dimfs^β(R)$. In the course of proving this, we also prove finite-state compression characterizations of $\dimfs^β(S)$ and $\Dimfs^β(S)$.
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