ON THE CONVERGENCE OF
DIRICHLET SERIES WITH RANDOM EXPONENTS
Andrii O. Kuryliak1, Oleh B. Skaskiv2, Nadiya Yu. Stasiv3
1,2,3Department of Mechanics and Mathematics
Ivan Franko National University of L'viv
Universytetska Street 1, L'viv - 79000, UKRAINE

Abstract

For the Dirichlet series of the form $\displaystyle F(z,\omega)=$ $=\sum\nolimits_{k=0}^{+\infty} f_k(\omega)e^{z\lambda_k(\omega)} \quad (z\in\mathbb{C},$ $\omega\in\Omega)$ with pairwise independent real exponents $(\lambda_k(\omega))$ on probability space $(\Omega,\mathcal{A},P)$ an estimates of abscissas convergence and absolutely convergence are established.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 30
Issue: 3
Year: 2017

DOI: 10.12732/ijam.v30i3.2

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References

  1. [1] S. Mandelbrojt, Séries de Dirichlet: Principes et Méthodes, Gauthier Villars, Paris (1969).
  2. [2] O.M. Mulyava, On the convergence abscissa of a Dirichlet series, Mat. Stud., 9, No 2 (1998), 171-176 (in Ukrainian).
  3. [3] O.Yu Zadorozhna, O.B. Skaskiv, Elementary remarks on the abscissas of the convergence of Laplace-Stieltjes integrals, Bukovyn. Mat. Zh., 1, No 3-4 (2013), 45-50 (in Ukrainian).
  4. [4] J.-P. Kahane, Some Random Series of Functions, 2nd Ed., Cambridge Univ. Press, Cambridge (1985).
  5. [5] Fanji Tian, Dao-chun Sun, Jia-rong Yu, Sur les séries aléatoires de Dirichlet, C. R. Acad. Sci. Paris, 326 (1998), 427-431.
  6. [6] Fanji Tian, Growth of random Dirichlet series, Acta Math. Sci., 20, No 3 (2000), 390-396.
  7. [7] H. Hedenmalm, Topics in the theory of Dirichlet series, Visn. Kharkiv. Un-tu. Ser. Mat., Prykl. Mat., Mekh., No 475 (2000), 195-203.
  8. [8] P.V. Filevych, On the relations between the abscissa of convergence and the abscissa of absolute convergence of random Dirichlet series, Mat. Stud., 20, No 1 (2003), 33-39.
  9. [9] X. Ding, Y. Xiao, Natural boundary of random Dirichlet series, Ukr. Mat. Zh., 58, No 7 (2006), 997-1005.
  10. [10] O.B. Skaskiv, L.O. Shapovalovska On the abscissas convergence random Dirichlet series, Bukovyn. Mat. Zh., 3, No 1 (2015), 110-114 (in Ukrainian).
  11. [11] H.J. Godwin, On generalizations of Tchebycheff's inequality, J. Amer Stat. Assoc., 50 (1955), 923-945.
  12. [12] I.R. Savage, Probability inequalities of the Tchebycheff type, J. of Research National Bureau of Standarts, 65B, No 3 (1961), 211-222.
  13. [13] P. Billingsley, Probability and Measure, Wiley, New York (1986).
  14. [14] C.D. Ryll-Nardzewski, Blackwell's conjecture on power series with random coefficients, Studia Math., 13 (1953), 30-36.
  15. [15] L. Arnold, Über die Konverge einer zuf¨alligen Potenzreihe, J. Reine Angew. Math., 222 (1966), 79-112.
  16. [16] L. Arnold, Konvergenzprobleme bei zufälligen Potenzreihen mit Lücken, Math. Zeitschr., 92 (1966), 356-365.
  17. [17] K. Roters, Convergence of random power series with pairwise independent Banach-space-valued coefficients, Statistics and Probability Letters, 18 (1993), 121-123.
  18. [18] L.O. Shapovalovska, O.B. Skaskiv, On the radius of convergence of random gap power series, Int. Journal of Math. Analysis, 9, No 38 (2015), 1889-1893.
  19. [19] Ph. Holgate, Some power series with random gaps, Adv. Appl. Prob., 21 (1989), 708-710.