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 | References | Similar Articles | Additional Information

Abstract: Generalizing classical studies of power series with sequences of independent random variables as coefficients, we study series of the forms

When is entire, its order of growth at infinity measures the speed of divergence of the ergodic averages of . We give examples to show that any order is possible for any and that different orders are possible for fixed . For fixed , the set of which produce infinite order is residual in the subset of consisting of those which are a.e. nonzero and produce entire . As in a theorem of Pólya for gap series, if is entire and has finite order, then it assumes every value infinitely many times.

The functions for which is rational a.e. are exactly the finite sums of eigenfunctions of ; 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 is rational, provided that takes one of several particular forms.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0854078-9

Keywords:
Random power series,
ergodic measure-preserving transformation,
order of growth,
lacunary series,
rational function

Article copyright:
© Copyright 1986
American Mathematical Society