Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
Full-text PDF Free Access

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

$\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.


References [Enhancements On Off] (What's this?)

  • [1] Jon Aaronson, An ergodic theorem with large normalising constants, Israel J. Math. 38 (1981), no. 3, 182–188. MR 605376, 10.1007/BF02760803
  • [2] J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. MR 0361090
  • [3] Hirotada Anzai, Ergodic skew product transformations on the torus, Osaka Math. J. 3 (1951), 83–99. MR 0040594
  • [4] Alexandra Ionescu Tulcea, Analytic continuation of random series, J. Math. Mech. 9 (1960), 399–410. MR 0113994
  • [5] Alexandra Ionescu Tulcea, Random series and spectra of measure-preserving transformations, Ergodic Theory (Proc. Internat. Sympos., Tulane Univ., New Orleans, La., 1961) Academic Press, New York, 1963, pp. 273–292. MR 0160873
  • [6] Judy D. Halchin, Analytic properties of random power series, Ph.D. dissertation, Univ. of North Carolina at Chapel Hill, 1983.
  • [7] R. E. A. C. Paley and A. Zygmund, On some series of functions (1), (2), (3), Proc. Cambridge Philos. Soc. 26 (1930), 337-357, 458-474; 28 (1932), 190-205.
  • [8] S. Saks and A. Zygmund, Analytic functions, Elsevier, New York, 1971.
  • [9] W. T. Sledd, Random series which are BMO or Bloch, Michigan Math. J. 28 (1981), no. 3, 259–266. MR 629359
  • [10] Hugo Steinhaus, Über die Wahrscheinlichkeit dafür, daß der Konvergenzkreis einer Potenzreihe ihre natürliche Grenze ist, Math. Z. 31 (1930), no. 1, 408–416 (German). MR 1545123, 10.1007/BF01246422
  • [11] -, Sur la probabilité de la convergence de séries, Studia Math. 2 (1930), 21-39.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 28D05, 30B10, 30D20

Retrieve articles in all journals with MSC: 28D05, 30B10, 30D20


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