Abstract
We establish a strong regularity property for the distributions of the random sums Σ±λn, known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (wn+1...,wn+ℓ) given the sum Σ ∞n=0 w n λn, tends to the uniform distribution on {±1}ℓ asn → ∞. More precise results, where ℓ grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems.
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The authors were partially supported by NSF grants DMS-9729992 (E. L.), DMS-9803597 (Y. P.) and DMS-0070538 (W. S.).
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Lindenstrauss, E., Peres, Y. & Schlag, W. Bernoulli convolutions and an intermediate value theorem for entropies ofK-partitions. J. Anal. Math. 87, 337–367 (2002). https://doi.org/10.1007/BF02868480
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DOI: https://doi.org/10.1007/BF02868480