Advertisement

Ukrainian Mathematical Journal

, Volume 58, Issue 7, pp 1129–1138 | Cite as

Natural boundary of random Dirichlet series

  • X. Ding
  • Y. Xiao
Article

Abstract

For the random Dirichlet series
$$\sum\limits_{n = 0}^\infty {X_n (\omega )e^{ - s\lambda _n } } (s = \sigma + it \in \mathbb{C}, 0 = \lambda _0 < \lambda _n \uparrow \infty )$$
whose coefficients are uniformly nondegenerate independent random variables, we provide some explicit conditions for the line of convergence to be its natural boundary a.s.

Keywords

Independent Random Variable Dirichlet Series Natural Boundary Martingale Difference Convergence Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.-P. Kahane, Some Random Series of Functions, Cambridge University Press, Cambridge (1985).zbMATHGoogle Scholar
  2. 2.
    C. D. Ryll-Nardzewski, “Blackwell’s conjecture on power series with random coefficients,” Stud. Math., 13, 30–36 (1953).MathSciNetzbMATHGoogle Scholar
  3. 3.
    P. Holgate, “The natural boundary problem for random power series with degenerate tail fields,” Ann. Probab., 11, 814–816 (1983).MathSciNetzbMATHGoogle Scholar
  4. 4.
    D. C. Su, “The natural boundary of some random power series,” Acta Math. Sci., 11, 463–470 (1991).Google Scholar
  5. 5.
    X. Q. Ding, “The singular points and Picard points of random Dirichlet series,” J. Math. (PRC), 18, 455–460 (1998).zbMATHGoogle Scholar
  6. 6.
    S. Mandelbrojt, Dirichlet Series. Principles and Methods, Reidel, Dordrecht (1972).zbMATHGoogle Scholar
  7. 7.
    J. R. Yu, Dirichlet Series and Random Dirichlet Series, Sci. Press, Beijing (1997).Google Scholar
  8. 8.
    D. L. Burkholder, “Independent sequence with the Stein property,” Ann. Math. Statist., 39, 1282–1288 (1968).MathSciNetzbMATHGoogle Scholar
  9. 9.
    Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, Springer, Berlin-New York (1978).zbMATHGoogle Scholar
  10. 10.
    J. Marcinkiewicz and A. Zygmund, “Sue les fonctions ind’ependantes,” Fund. Math., 29, 60–90 (1937).Google Scholar
  11. 11.
    R. F. Gundy, “The martingale version of a theorem of Marcinkiewicz and Zygmund,” Ann. Math. Statist., 38, 725–734 (1967).MathSciNetzbMATHGoogle Scholar
  12. 12.
    W. F. Stout, Almost Sure Convergence, Academic Press, New York (1974).zbMATHGoogle Scholar
  13. 13.
    R. E. A. C. Paley and A. Zygmund, “On some series of functions (3),” Proc. Cambridge Phil. Soc., 28, 190–205 (1932).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • X. Ding
    • 1
  • Y. Xiao
    • 2
  1. 1.Northwestern Polytechnical UniversityXi’anChina
  2. 2.Michigan State UniversityEast LansingUSA

Personalised recommendations