Abstract
If $T$ is an automorphism on a Lebesgue space and $P$ a finite generator for $T$, then $T$ is a Bernoulli shift if $$\sup \{|\mu(A \cap B) - \mu(A)\mu(B)|: A \in \vee^{-1}_{-\infty} T^j P, B \in \vee^\infty_k T^j P\}$$ is $o(|P|^{-a}k)$ where $a_k/k \rightarrow \infty$ as $k \rightarrow \infty$.
Citation
N. F. G. Martin. "The Rosenblatt Mixing Condition and Bernoulli Shifts." Ann. Probab. 2 (2) 333 - 338, April, 1974. https://doi.org/10.1214/aop/1176996715
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