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Bernoulli convolutions and an intermediate value theorem for entropies ofK-partitions

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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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Authors and Affiliations

  1. Institute for Advanced Study, Olden Lane, 08540, Princeton, NJ, USA

    Elon Lindenstrauss

  2. Department of Statistics, University of California, 94720, Berkeley, CA, USA

    Yuval Peres

  3. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Yuval Peres

  4. Department of Mathematics, Princeton University, Fine Hall, 08544, Princeton, NJ, USA

    Wilhelm Schlag

Authors
  1. Elon Lindenstrauss
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  2. Yuval Peres
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  3. Wilhelm Schlag
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Corresponding author

Correspondence to Elon Lindenstrauss.

Additional information

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

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