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

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

MathSciNet review:
854078

Full-text PDF

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.

**[1]**Jon Aaronson,*An ergodic theorem with large normalizing constants*, Israel J. Math.**38**(1981), 182-188. MR**605376 (83f:28014)****[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 (50:13536)****[3]**Hirotada Anzai,*Ergodic skew product transformations on the torus*, Osaka Math. J.**3**(1951), 83-99. MR**0040594 (12:719d)****[4]**Alexandra Ionescu Tulcea,*Analytic continuation of random series*, J. Math. Mech.**9**(1960), 399-410. MR**0113994 (22:4825)****[5]**-,*Random series and spectra of measure-preserving transformations*, Ergodic Theory (Fred B. Wright, ed.), Academic Press, New York, 1963, pp. 273-292. MR**0160873 (28:4082)****[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**or Bloch*, Michigan Math. J.**28**(1981), 259-266. MR**629359 (82k:30004)****[10]**Hugo Steinhaus,*Über die Wahrscheinlichkeit dafür, dass der Konvergenzkreis einer Potenzreihe ihre natürliche Grenze ist*, Math. Z.**31**(1929), 408-416. MR**1545123****[11]**-,*Sur la probabilité de la convergence de séries*, Studia Math.**2**(1930), 21-39.

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