Subnormal independent random variables and Levy’s phenomenon for entire functions

A. Kuryliak

doi:10.15330/ms.47.1.10-19

Suppose that $(Z_n)$ is a sequence of real independent subnormal random variables, i.e. such that there exists $D>0$ satisfying following inequality for expectation $ \mathbf{E}(e^{\lambda_0Z_k})\leq e ^{D \lambda_0^2} $ for any $k\in\mathbb{N}$ for all $\lambda_0\in\mathbb{R}$. In this paper is proved that for random entire functions of the form $ f(z,\omega)=\sum_{n=0}^{+\infty}Z_n(\omega)a_nz^n $ Levy's phenomenon holds.

1. G. Valiron, Fonctions analytiques. - Paris: Press. Univer. de France, 1954.

2. H. Wittich, Neuere Untersuchungen uber eindeutige analytische Funktionen, Springer-Verlag, Berlin- Gottingen-Heidelberg, 1955.

3. G. Polya, G. Szego, Aufgaben und Lehrsatze aus der Analysis, V.2. - Berlin, Springer, 1925.

4. O.B. Skaskiv, P.V. Filevych, On the size of an exceptional set in the Wiman theorem, Mat. Stud., 12 (1999), №1, 31-36. (in Ukrainian)

5. M.M. Sheremeta, Wiman-Valirons method for entire functions, represented by Dirichlet series, Dokl. USSR Acad. Sci., 240 (1978), №5, 1036-1039. (in Russian)

6. O.B. Skaskiv, On the classical Wimans inequality for entire Dirichlet series, Visn. Lviv. Univer., ser. mekh.-mat., 54 (1999), 180-182. (in Ukrainian)

7. O.B. Skaskiv, Random gap power series and Wimans inequality, Mat. Stud. 30 (2008), №1, 101-106. (in Ukrainian)

8. P. Levy, Sur la croissante de fonctions entiere, Bull. Soc. Math. France, 58 (1930), 29-59; 127-149.

9. P. Erd.os, A. Renyi On random entire functions, Zastos. Mat., 10 (1969), 47-55.

10. P.V. Filevych, Correlation between the maximum modulus and maximal term of random entire functions, Mat. Stud., 7 (1997), №2, 157-166.

11. O.B. Skaskiv, O.V. Zrum, On an exeptional set in the Wiman inequalities for entire functions, Mat. Stud., 21 (2004), №1, 13-24. (in Ukrainian)

12. J.P. Kahane, Some random series of functions, Cambridge University Press, 1985, 305 p.

13. O.B. Skaskiv, O.V. Zrum, Refinemant of Fentons inequality for entire functions of two complex variables, Mat. Visn. Nauk. Tov. Im. Shevchenka, 3 (2006), 56-68. (in Ukrainian)

14. O.B. Skaskiv, O.V. Zrum, Wiman-type inequalities for entire functions of two complex variables with rapidly oscillating coefficients, Mat. Metody Phys.-Mekh. Polya, 48 (2005), №4, 78-87. (in Ukrainian)

15. O.V. Zrum, O.B. Skaskiv, On Wimans inequality for random entire functions of two variables, Mat. Stud., 23 (2005), №2, 149-160. (in Ukrainian)

16. A.O. Kuryliak, O.B. Skaskiv, O.V. Zrum, Levys phenomenon for entire functions of several variables, Ufa Math. Journ., 6 (2014), №2, 118-127.

17. A.O. Kuryliak, L.O. Shapovalovska, Wimans type inequality for entire functions of several complex variables with rapidly oscillating coefficients, Mat. Stud., 43 (2015), №1, 16-26.

18. J.M. Steele, Wimans inequality for entire functions with rapidly oscilating coefficients, J. Math. Anal. Appl., 123 (1987), 550-558.

19. P. Filevych, Some classes of entire functions in which the Wiman-Valiron inequality can be almost certainly improved, Mat. Stud., 6 (1996), 59-66. (in Ukrainian)

20. F. Tian, Growth of random Dirichlet series, Acta Math. Sc., 20B (2000), №3, 390-396.

21. L.O. Shapovalovska, O.B. Skaskiv, On the radius of convergence of random gap power series, Int. Journal of Math. Analysis, 9 (2015), №38, 1889-1893.

22. W.K. Hayman, Subharmonic functions, V.2, Academic Press, London, 1989.

23. A. Nishry, The hole probability for gaussian entire functions, arxiv:0909.12v3, 2009.

24. W.K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull., 17 (1974), №3, 317-358.

25. R. Salem, A. Zygmund, Some properties of trigonometric series whose terms have random sign, Acta. Math., 91 (1954), 245-301.

	Subnormal independent random variables and Levy’s phenomenon for entire functions
Author	A. Kuryliak andriykuryliak@gmail.com Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	Suppose that $(Z_n)$ is a sequence of real independent subnormal random variables, i.e. such that there exists $D>0$ satisfying following inequality for expectation $ \mathbf{E}(e^{\lambda_0Z_k})\leq e ^{D \lambda_0^2} $ for any $k\in\mathbb{N}$ for all $\lambda_0\in\mathbb{R}$. In this paper is proved that for random entire functions of the form $ f(z,\omega)=\sum_{n=0}^{+\infty}Z_n(\omega)a_nz^n $ Levy's phenomenon holds.
Keywords	entire function; Levy’s phenomenon; subnormal random variables
DOI	doi:10.15330/ms.47.1.10-19
Reference	1. G. Valiron, Fonctions analytiques. - Paris: Press. Univer. de France, 1954. 2. H. Wittich, Neuere Untersuchungen uber eindeutige analytische Funktionen, Springer-Verlag, Berlin- Gottingen-Heidelberg, 1955. 3. G. Polya, G. Szego, Aufgaben und Lehrsatze aus der Analysis, V.2. - Berlin, Springer, 1925. 4. O.B. Skaskiv, P.V. Filevych, On the size of an exceptional set in the Wiman theorem, Mat. Stud., 12 (1999), №1, 31-36. (in Ukrainian) 5. M.M. Sheremeta, Wiman-Valirons method for entire functions, represented by Dirichlet series, Dokl. USSR Acad. Sci., 240 (1978), №5, 1036-1039. (in Russian) 6. O.B. Skaskiv, On the classical Wimans inequality for entire Dirichlet series, Visn. Lviv. Univer., ser. mekh.-mat., 54 (1999), 180-182. (in Ukrainian) 7. O.B. Skaskiv, Random gap power series and Wimans inequality, Mat. Stud. 30 (2008), №1, 101-106. (in Ukrainian) 8. P. Levy, Sur la croissante de fonctions entiere, Bull. Soc. Math. France, 58 (1930), 29-59; 127-149. 9. P. Erd.os, A. Renyi On random entire functions, Zastos. Mat., 10 (1969), 47-55. 10. P.V. Filevych, Correlation between the maximum modulus and maximal term of random entire functions, Mat. Stud., 7 (1997), №2, 157-166. 11. O.B. Skaskiv, O.V. Zrum, On an exeptional set in the Wiman inequalities for entire functions, Mat. Stud., 21 (2004), №1, 13-24. (in Ukrainian) 12. J.P. Kahane, Some random series of functions, Cambridge University Press, 1985, 305 p. 13. O.B. Skaskiv, O.V. Zrum, Refinemant of Fentons inequality for entire functions of two complex variables, Mat. Visn. Nauk. Tov. Im. Shevchenka, 3 (2006), 56-68. (in Ukrainian) 14. O.B. Skaskiv, O.V. Zrum, Wiman-type inequalities for entire functions of two complex variables with rapidly oscillating coefficients, Mat. Metody Phys.-Mekh. Polya, 48 (2005), №4, 78-87. (in Ukrainian) 15. O.V. Zrum, O.B. Skaskiv, On Wimans inequality for random entire functions of two variables, Mat. Stud., 23 (2005), №2, 149-160. (in Ukrainian) 16. A.O. Kuryliak, O.B. Skaskiv, O.V. Zrum, Levys phenomenon for entire functions of several variables, Ufa Math. Journ., 6 (2014), №2, 118-127. 17. A.O. Kuryliak, L.O. Shapovalovska, Wimans type inequality for entire functions of several complex variables with rapidly oscillating coefficients, Mat. Stud., 43 (2015), №1, 16-26. 18. J.M. Steele, Wimans inequality for entire functions with rapidly oscilating coefficients, J. Math. Anal. Appl., 123 (1987), 550-558. 19. P. Filevych, Some classes of entire functions in which the Wiman-Valiron inequality can be almost certainly improved, Mat. Stud., 6 (1996), 59-66. (in Ukrainian) 20. F. Tian, Growth of random Dirichlet series, Acta Math. Sc., 20B (2000), №3, 390-396. 21. L.O. Shapovalovska, O.B. Skaskiv, On the radius of convergence of random gap power series, Int. Journal of Math. Analysis, 9 (2015), №38, 1889-1893. 22. W.K. Hayman, Subharmonic functions, V.2, Academic Press, London, 1989. 23. A. Nishry, The hole probability for gaussian entire functions, arxiv:0909.12v3, 2009. 24. W.K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull., 17 (1974), №3, 317-358. 25. R. Salem, A. Zygmund, Some properties of trigonometric series whose terms have random sign, Acta. Math., 91 (1954), 245-301.
Pages	10-19
Volume	47
Issue	1
Year	2017
Journal	Matematychni Studii
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Table of content of issue	html