Skip to main content
SpringerLink
Log in

Natural boundary of the random power series

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We investigate convergence properties of random Taylor series whose coefficients are ψ-mixing random variables. In particular, we give sufficient conditions such that the circle of the convergence of the series forms almost surely a natural boundary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Annoni M, Grafakos L. On an inequality of Sagher and Zhou concerning Stein’s lemma. Collect Math, 2009, 60: 297–306

    Article  MathSciNet  Google Scholar 

  2. Astashkin S V. Khintchine inequality for sets of small measure. Funct Anal Appl, 2014, 48: 235–241

    Article  MathSciNet  Google Scholar 

  3. Astashkin S V, Curbera G P. Local Khintchine inequality in rearrangement invariant spaces. Ann Mat Pura Appl (4), 2015, 194: 619–643

    Article  MathSciNet  Google Scholar 

  4. Bhowmik G, Matsumoto K. Analytic continuation of random Dirichlet series. Proc Steklov Inst Math, 2013, 282: S67–S72

    Article  MathSciNet  Google Scholar 

  5. Blum J R, Hanson D L, Koopmans L H. On the strong law of large numbers for a class of stochastic processes. Z Wahrsch Verw Gebiete, 1963, 2: 1–11

    Article  MathSciNet  Google Scholar 

  6. Borel E. Sur les series de Taylor. C R Math Acad Sci Paris, 1896, 123: 1051–1052

    MATH  Google Scholar 

  7. Bradley R C. Basic properties of strong mixing conditions. A survey and some open questions. Probab Surv, 2005, 2: 107–144

    Article  MathSciNet  Google Scholar 

  8. Breuer J, Simon B. Natural boundaries and spectral theory. Adv Math, 2011, 226: 4902–4920

    Article  MathSciNet  Google Scholar 

  9. Burkholder D L. Independent sequences with the Stein property. Ann Math Statist, 1968, 39: 1282–1288

    Article  MathSciNet  Google Scholar 

  10. Burkholder D L. The 1971 wald memorial lectures: Distribution function inequalities for martingales. Ann Probab, 1973, 1: 19–42

    Google Scholar 

  11. Carrillo-Alanís J. On local Khintchine inequalities for spaces of exponential integrability. Proc Amer Math Soc, 2011, 139: 2753–2757

    Article  MathSciNet  Google Scholar 

  12. Ding X Q, Xiao Y M. Natural boundary of random Dirichlet series. Ukrainian Math J, 2006, 58: 1129–1138

    Article  MathSciNet  Google Scholar 

  13. Durrett R. Probability: Theory and Examples. Cambridge: Cambridge University Press, 2010

    Book  Google Scholar 

  14. Grafakos L. Classical Fourier Analysis. New York: Springer-Verlag, 2008

    MATH  Google Scholar 

  15. Heil C. A Basis Theory Primer, expanded ed. New York: Birkhäuser/Springer, 2011

    Book  Google Scholar 

  16. Holgate P. The natural boundary problem for random power series with degenerate tail fields. Ann Probab, 1983, 11: 814–816

    Article  MathSciNet  Google Scholar 

  17. Ionescu Tulcea A. Analytic continuation of random series. J Math Mech, 1960, 9: 399–410

    MathSciNet  MATH  Google Scholar 

  18. Kahane J P. Some Random Series of Function. Cambridge: Cambridge University Press, 1985

    MATH  Google Scholar 

  19. Kochen S, Stone C. A note on the Borel-Cantelli lemma. Illinois J Math, 1964, 8: 248–251

    Article  MathSciNet  Google Scholar 

  20. Ledoux M, Talagrand M. Probability in Banach Space. Berlin: Springer, 1991

    Book  Google Scholar 

  21. Lin Z Y, Lu C R. Limit Theory for Mixing and Dependent Random Variables. Dordrecht: Kluwer Academic Publishers; Beijing: Science Press, 1997

    Google Scholar 

  22. Marcinkiewicz J, Zygmund A. Sue les fonctions indépendante. Fund Math Statist, 1967, 28: 190–205

    Google Scholar 

  23. Paley R E A C, Zygmund A. A note on analytic functions in the unit circle. Math Proc Cambridge Philos Soc, 1932, 28: 266–272

    Article  Google Scholar 

  24. Philipp W. The central limit problem for mixing sequences of random variables. Z Wahrsch Verw Gebiete, 1969, 12: 155–171

    Article  MathSciNet  Google Scholar 

  25. Ryll-Nardzewski C. D Blackwell’s conjecture on power series with random coefficients. Studia Math, 1953, 13: 30–36

    Article  MathSciNet  Google Scholar 

  26. Sagher Y, Zhou K C. Local norm inequalities for lacunary series. Indiana Univ Math J, 1990, 39: 45–60

    Article  MathSciNet  Google Scholar 

  27. Stein E M. On limits of sequences of operators. Ann of Math (2), 1961, 74: 140–170

    Article  MathSciNet  Google Scholar 

  28. Steinhaus H. Uber dic Wahrscheilicheit dafür, dass der Konvergenzkeis einer Potenzreiheihre natürlich Grenze ist. Math Z, 1930, 31: 408–416

    Article  MathSciNet  Google Scholar 

  29. Sun D C. The natural boundary of some random power series. Acta Math Sci Ser B Engl Ed, 1991, 11: 463–470

    Article  MathSciNet  Google Scholar 

  30. Sun D C, Yu J R. Sur la distribution des valurs de certaines series aléatoires de Dirichlet. II. C R Acad Sci Sér I Math, 1989, 308: 205–207

    MATH  Google Scholar 

  31. Yan J A. A simple proof of two generalized Borel-Cantelli lemmas. In: Memoriam Paul-André Meyer. Lecture Notes in Mathematics, vol. 1874. Berlin-Heidelberg: Springer, 2006, 77–79

    Chapter  Google Scholar 

  32. Yu J R. Julia lines of random Dirichlet series. Bull Sci Math, 2004, 128: 341–353

    Article  MathSciNet  Google Scholar 

  33. Zygmund A. On the convergence of lacunary trigonometric series. Fund Math, 1930, 16: 90–107

    Article  Google Scholar 

  34. Zygmund A. Trigonometric Series. Cambridge: Cambridge University Press, 1977

    MATH  Google Scholar 

Download references

Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11501127). The third author was supported by the Luxembourg National Research Fund (FNR) (Grant No. R-AGR-3410-12-Z). The fourth author was supported by National Natural Science Foundation of China (Grant No. 11801591) and Science and Technology Program of Guangzhou (Grant No. 202002030369). The first author expresses her gratitude to Professor Yimin Xiao for giving her a chance to visit the Department of Statistics and Probability in Michigan State University in 2017, where part of this work was conducted. At last, the authors are greatly indebted to the referees for suggesting several improvements to this paper.

Author information

Authors and Affiliations

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510520, China

    Yingying Huo

  2. School of Mathematics, South China Normal University, Guangzhou, 510631, China

    Daochun Sun

  3. Mathematics Research Unit, Université du Luxembourg, Esch-sur-Alzette, L-4364, Luxembourg

    Xiaochuan Yang

  4. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Xiaochuan Yang

  5. School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China

    Lulu Fang

Authors
  1. Yingying Huo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daochun Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaochuan Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lulu Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lulu Fang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huo, Y., Sun, D., Yang, X. et al. Natural boundary of the random power series. Sci. China Math. 65, 951–970 (2022). https://doi.org/10.1007/s11425-019-1765-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-1765-1

Keywords

MSC(2020)

Search

Navigation