A Notion of Entropy for Stochastic Processes

The paper defines Hsd(µ) via limsup growth of block-partition entropies and proves, for any stationary measure µ on X^∞, that Hsd(µ) equals the Kolmogorov–Sinai entropy of the one-sided shift E(Sh,µ) (Theorems 3.2 and 3.3) . The candidate solution establishes the same equality by a standard entropy-calculus route: (i) identify Hsd(µ,P) with hµ(Sh,αP) for zero-coordinate cylinders, yielding Hsd(µ)=supP hµ(Sh,αP)≤hµ(Sh); and (ii) approximate arbitrary partitions by finite-block partitions and then by cylinder-joins from a single base partition P, concluding the reverse inequality. The paper’s proof appeals to Walters’ generator/supremum theorem to reduce to finite-block partitions directly, while the model gives a more constructive approximation using conditional entropies. Both arguments are correct and consistent with standard results; they differ in technique (generator-based reduction versus stepwise conditional-entropy approximation) but reach the same conclusion .