Bernoulli convolutions and an intermediate value theorem for entropies of<Emphasis Type="Italic">K</Emphasis>-partitions |
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Authors: | Email author" target="_blank">Elon?LindenstraussEmail author Yuval?Peres Wilhelm?Schlag |
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Institution: | (1) Institute for Advanced Study, Olden Lane, 08540 Princeton, NJ, USA;(2) Department of Statistics, University of California, 94720 Berkeley, CA, USA;(3) Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel;(4) Department of Mathematics, Princeton University, Fine Hall, 08544 Princeton, NJ, USA |
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Abstract: | We establish a strong regularity property for the distributions of the random sums Σ±λ
n
, known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (w
n+1...,w
n+ℓ) given the sum Σ
n=0
∞
w
n
λ
n
, 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.
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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Keywords: | |
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