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Transactions of the American Mathematical Society

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Random power series generated by ergodic transformations

Authors: Judy Halchin and Karl Petersen
Journal: Trans. Amer. Math. Soc. 297 (1986), 461-485
MSC: Primary 28D05; Secondary 30B10, 30D20
MathSciNet review: 854078
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Abstract: Generalizing classical studies of power series with sequences of independent random variables as coefficients, we study series of the forms

$\displaystyle {g_{x,\phi }}(z) = \sum\limits_{n = 0}^\infty {\phi ({T^n}x){z^n}... ...m\limits_{n = 1}^\infty {\phi (x)\phi (Tx) \cdots \phi ({T^{n - 1}}x){z^n},} } $

where $ T$ is an ergodic measure-preserving transformation on a probability space $ (X,\mathcal{B},\mu )$ and $ \phi $ is a measurable complex-valued function which is a.e. nonzero.

When $ {f_{x,\phi }}$ is entire, its order of growth at infinity measures the speed of divergence of the ergodic averages of $ \log \vert\phi \vert$. We give examples to show that any order is possible for any $ T$ and that different orders are possible for fixed $ \phi $. For fixed $ T$, the set of $ \phi $ which produce infinite order is residual in the subset of $ {L^1}(X)$ consisting of those $ \phi $ which are a.e. nonzero and produce entire $ {f_{x,\phi }}$. As in a theorem of Pólya for gap series, if $ {f_{x,\phi }}$ is entire and has finite order, then it assumes every value infinitely many times.

The functions $ \phi \in {L^1}(X)$ for which $ {g_{x,\phi }}$ is rational a.e. are exactly the finite sums of eigenfunctions of $ T$; their poles are all simple and are the inverses of the corresponding eigenvalues. By combining this result with a skew product construction, we can also characterize when $ {f_{x,\phi }}$ is rational, provided that $ \phi $ takes one of several particular forms.

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Keywords: Random power series, ergodic measure-preserving transformation, order of growth, lacunary series, rational function
Article copyright: © Copyright 1986 American Mathematical Society

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