On convergence of random multiple Dirichlet series

A. O. Kuryliak; O. B. Skaskiv; N. Yu. Stasiv

doi:10.15330/ms.49.2.122-137

Let

$F_{\omega}(s)=\sum_{\|n\|=0}^\infty f_{(n)}(\omega)\exp\{(\lambda_{(n)},s)\},$ \ where the exponents

$\lambda_{(n)}=(\lambda_{n_1}^{(1)},\ldots,\lambda_{n_p}^{(p)})\in\mathbb{R}^p_+$ ,

$(n)=(n_1,\ldots,n_p)\in\mathbb{Z}^p_+$ ,

$p\in\mathbb{N}$ ,

$\|n\|=n_1+\ldots+n_p$ , and the coefficients

$f_{(n)}(\omega)$ are pairwise independent random complex variables. In the paper, in particular, we prove the following statements: 1) If

$\tau(\lambda)=\varlimsup_{\|n\|\to +\infty}\ln \|n\|/\|\lambda_{(n)}\|=0$ , then in order that a Dirichlet series be convergent a.s. in the whole space

$\mathbb{C}^p$ , it is necessary and sufficient that

$(\forall \Delta>0)\colon\ \sum\nolimits_{\|n\|=0}^{+\infty}\big(1-F_{(n)}\big(\exp(-\Delta\|\lambda_{(n)}\|)\big)\big)\le+\infty.$ 2) If

$\tau(\lambda)=0$ , then in order that

$\sigma\in \partial G_{a}\cap\big(\mathbb{R}_+\setminus\{0\}\big)^p$ a.s., it is necessary and sufficient that

$(\forall \varepsilon>0)\colon\ \sum_{\|n\|=0}^{+\infty}\Big(1-F_{(n)}\big(e^{(-1+\varepsilon)(\sigma,\lambda_{(n)})}\big)\Big)\le+\infty\ \wedge\ \sum_{\|n\|=0}^{+\infty}\Big(1-F_{(n)}\big(e^{(-1-\varepsilon)(\sigma,\lambda_{(n)})}\big)\Big)=+\infty,$ where

$F_{(n)}(x):=P\{ \omega : |f_{(n)}(\omega)|\le x\},\ x\in \mathbb R,\ (n)\in\mathbb{Z}^p_+$ is the distribution function of

$|f_{(n)}(\omega)|$ ,

$\partial G_a$ is the set of conjugate abscissas of absolute convergence of the random Dirichlet series

$F_{\omega}$ .

1. T. Kojima, On the double Dirichlets series, Sc. Reports Tohoku Imperial Univ. (1), Math., Phys., Chem., 9 (1920), 351-340.

2. V.P. Gromov, Multiple series of Dirichlet polynomials, Sib. Math. J., 10 (1969), №3, 374-386, English transl. from Sibirski Matematicheski Zhurnal, 10 (1969), №3, 522-536.

3. V.P. Gromov, To the theory of multiple Dirichlet series, Izv. AN ArmSSR. Mat., 5 (1970), №5, 449-457. (in Russian)

4. V.P. Gromov, To the theory of multiple Dirichlet series, Izv. AN ArmSSR. Mat., 7 (1970), №2, 90-103. (in Russian)

5. A.I. Janusauskas, Double Dirichlet series, Lit. Mat. Sb., 18 (1978), №3, 201.211, English transl. in Lit. Math. J., 18 (1978), №3, 445-452.

6. A.I. Janusauskas, Properties of conjugate abscissas of convergence of double Dirichleet series, Lit. Mat. Sb., 19 (1979), №1, 213.228, English transl. in Lit. Math. J. 19 (1979), №1, 152-164.

7. O.Yu. Zadorozhna, O.M. Mulyava, On the conjugate abscissas convergence of the double Dirichlet series, Mat. Stud., 28 (2007), №1, 29-35. (in Ukrainian)

8. O.Yu. Zadorozhna, O.B. Skaskiv, On the conjugate abscissas convergence of multiple Dirichlet series, Carpathian Mathematical Publications, 1 (2009), №2, 152.160. (in Ukrainian)

9. O.Yu. Zadorozhna, O.B. Skaskiv, On the domains of convergence of the double Dirichlet series, Mat. Stud., 32 (2009), №1, 81-86. (in Ukrainian)

10. O.B. Skaskiv, O.Yu. Zadorozhna, On domains of convergence of multiple random Dirichlet series, Mat. Stud., 36 (2011), №1, 51-57.

11. A.O. Kuryliak, O.B. Skaskiv, N.Yu. Stasiv, Abscissas convergence of the random multiple Dirichlet series, Precarpatian visn. NTSh. Chyslo, (2018), №1, (42), 1-11. (in Ukrainian)

12. O.M. Mulyava, On the convergence abscissa of a Dirichlet series, Mat. Stud., 9 (1998), №2, 171-176. (in Ukrainian)

13. O.Yu. Zadorozhna, O.B. Skaskiv, Elementary remarks about abscissas of convergence of the Laplace- Stieltjes integrals, Bukovyn. mat. zhurn., 1 (2013), №3-4, 45-50. (in Ukrainian)

14. O.B. Skaskiv, A.I. Bandura, Asymptotic estimates of positive integrals and entire functions, Lviv-Ivano-Frankivsk: Goliney, 2015, 108 p.

15. Skaskiv O.B., Stasiv N.Yu. Abscissas of convergence of Dirichlet series with random exponents, Visn. Lviv. Univ. Ser. mech.-mat., 84 (2017), 76-91. (in Ukrainian)

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

17. Skaskiv O.B., Shapovalovska L.O. On the abscissas of convergence of random Dirichlet series, Bukovyn. Mat. Zhurn., 3 (2015), №1, 110-114. (in Ukrainian)

18. Kuryliak A.O., Skaskiv O.B., Stasiv O.Yu. On the abscissas of convergence of Dirichlet series with random pairwise independent exponents, ArXiv:1703.03280v1[math.CV] 12 Mar 2017, 12 p.

19. Kuryliak A.O., Skaskiv O.B., Stasiv O.Yu. On the convergence of Dirichlet series with random exponents, Int. Journal of Appl. Math., 30 (2017), №3, 229-238.

20. Kuryliak A.O., Skaskiv O.B., Stasiv N.Yu. On the abscissas of convergence of Dirichlet series with the random exponents and coefficients, Bukovyn. Mat. Zhurn., 5 (2017), №3-4, 90-97.

21. Erdos P., Renyi A. On Cantors series with convergent

$\sum 1/q_n$ , Ann. Univ. Sci. Budapest Eotvos. Sect. Math., 2 (1959), 93-109.

22. Petrov V.V., Sums of idependent random variables, New York: Springer, 1975.

23. Billingsley P., Probability and measure, New York: Wiley, 1986.

	On convergence of random multiple Dirichlet series
Author	A. O. Kuryliak¹, O. B. Skaskiv², N. Yu. Stasiv³ andriykuryliak@gmail.com¹, olskask@gmail.com², n-stas@ukr.net³ 1) Ivan Franko National University of L’viv, Lviv, Ukraine; 2) Ivan Franko National University of L’viv, Lviv, Ukraine; 3) Ivan Franko National University of L’viv, Lviv, Ukraine
Abstract	Let $F_{\omega}(s)=\sum_{\\|n\\|=0}^\infty f_{(n)}(\omega)\exp\{(\lambda_{(n)},s)\},$ \ where the exponents $\lambda_{(n)}=(\lambda_{n_1}^{(1)},\ldots,\lambda_{n_p}^{(p)})\in\mathbb{R}^p_+$ , $(n)=(n_1,\ldots,n_p)\in\mathbb{Z}^p_+$ , $p\in\mathbb{N}$ , $\\|n\\|=n_1+\ldots+n_p$ , and the coefficients $f_{(n)}(\omega)$ are pairwise independent random complex variables. In the paper, in particular, we prove the following statements: 1) If $\tau(\lambda)=\varlimsup_{\\|n\\|\to +\infty}\ln \\|n\\|/\\|\lambda_{(n)}\\|=0$ , then in order that a Dirichlet series be convergent a.s. in the whole space $\mathbb{C}^p$ , it is necessary and sufficient that $(\forall \Delta>0)\colon\ \sum\nolimits_{\\|n\\|=0}^{+\infty}\big(1-F_{(n)}\big(\exp(-\Delta\\|\lambda_{(n)}\\|)\big)\big)\le+\infty.$ 2) If $\tau(\lambda)=0$ , then in order that $\sigma\in \partial G_{a}\cap\big(\mathbb{R}_+\setminus\{0\}\big)^p$ a.s., it is necessary and sufficient that $(\forall \varepsilon>0)\colon\ \sum_{\\|n\\|=0}^{+\infty}\Big(1-F_{(n)}\big(e^{(-1+\varepsilon)(\sigma,\lambda_{(n)})}\big)\Big)\le+\infty\ \wedge\ \sum_{\\|n\\|=0}^{+\infty}\Big(1-F_{(n)}\big(e^{(-1-\varepsilon)(\sigma,\lambda_{(n)})}\big)\Big)=+\infty,$ where $F_{(n)}(x):=P\{ \omega : \|f_{(n)}(\omega)\|\le x\},\ x\in \mathbb R,\ (n)\in\mathbb{Z}^p_+$ is the distribution function of $\|f_{(n)}(\omega)\|$ , $\partial G_a$ is the set of conjugate abscissas of absolute convergence of the random Dirichlet series $F_{\omega}$ .
Keywords	multiple Dirichlet series; conjugate abscissas of convergence; random exponents
DOI	doi:10.15330/ms.49.2.122-137
Reference	1. T. Kojima, On the double Dirichlets series, Sc. Reports Tohoku Imperial Univ. (1), Math., Phys., Chem., 9 (1920), 351-340. 2. V.P. Gromov, Multiple series of Dirichlet polynomials, Sib. Math. J., 10 (1969), №3, 374-386, English transl. from Sibirski Matematicheski Zhurnal, 10 (1969), №3, 522-536. 3. V.P. Gromov, To the theory of multiple Dirichlet series, Izv. AN ArmSSR. Mat., 5 (1970), №5, 449-457. (in Russian) 4. V.P. Gromov, To the theory of multiple Dirichlet series, Izv. AN ArmSSR. Mat., 7 (1970), №2, 90-103. (in Russian) 5. A.I. Janusauskas, Double Dirichlet series, Lit. Mat. Sb., 18 (1978), №3, 201.211, English transl. in Lit. Math. J., 18 (1978), №3, 445-452. 6. A.I. Janusauskas, Properties of conjugate abscissas of convergence of double Dirichleet series, Lit. Mat. Sb., 19 (1979), №1, 213.228, English transl. in Lit. Math. J. 19 (1979), №1, 152-164. 7. O.Yu. Zadorozhna, O.M. Mulyava, On the conjugate abscissas convergence of the double Dirichlet series, Mat. Stud., 28 (2007), №1, 29-35. (in Ukrainian) 8. O.Yu. Zadorozhna, O.B. Skaskiv, On the conjugate abscissas convergence of multiple Dirichlet series, Carpathian Mathematical Publications, 1 (2009), №2, 152.160. (in Ukrainian) 9. O.Yu. Zadorozhna, O.B. Skaskiv, On the domains of convergence of the double Dirichlet series, Mat. Stud., 32 (2009), №1, 81-86. (in Ukrainian) 10. O.B. Skaskiv, O.Yu. Zadorozhna, On domains of convergence of multiple random Dirichlet series, Mat. Stud., 36 (2011), №1, 51-57. 11. A.O. Kuryliak, O.B. Skaskiv, N.Yu. Stasiv, Abscissas convergence of the random multiple Dirichlet series, Precarpatian visn. NTSh. Chyslo, (2018), №1, (42), 1-11. (in Ukrainian) 12. O.M. Mulyava, On the convergence abscissa of a Dirichlet series, Mat. Stud., 9 (1998), №2, 171-176. (in Ukrainian) 13. O.Yu. Zadorozhna, O.B. Skaskiv, Elementary remarks about abscissas of convergence of the Laplace- Stieltjes integrals, Bukovyn. mat. zhurn., 1 (2013), №3-4, 45-50. (in Ukrainian) 14. O.B. Skaskiv, A.I. Bandura, Asymptotic estimates of positive integrals and entire functions, Lviv-Ivano-Frankivsk: Goliney, 2015, 108 p. 15. Skaskiv O.B., Stasiv N.Yu. Abscissas of convergence of Dirichlet series with random exponents, Visn. Lviv. Univ. Ser. mech.-mat., 84 (2017), 76-91. (in Ukrainian) 16. Shapovalovska L.O., Skaskiv O.B. On the radius of convergence of random gap power series, Int. Journal of Math. Analysis, 9 (2015), №38, 1889-1893. 17. Skaskiv O.B., Shapovalovska L.O. On the abscissas of convergence of random Dirichlet series, Bukovyn. Mat. Zhurn., 3 (2015), №1, 110-114. (in Ukrainian) 18. Kuryliak A.O., Skaskiv O.B., Stasiv O.Yu. On the abscissas of convergence of Dirichlet series with random pairwise independent exponents, ArXiv:1703.03280v1[math.CV] 12 Mar 2017, 12 p. 19. Kuryliak A.O., Skaskiv O.B., Stasiv O.Yu. On the convergence of Dirichlet series with random exponents, Int. Journal of Appl. Math., 30 (2017), №3, 229-238. 20. Kuryliak A.O., Skaskiv O.B., Stasiv N.Yu. On the abscissas of convergence of Dirichlet series with the random exponents and coefficients, Bukovyn. Mat. Zhurn., 5 (2017), №3-4, 90-97. 21. Erdos P., Renyi A. On Cantors series with convergent $\sum 1/q_n$ , Ann. Univ. Sci. Budapest Eotvos. Sect. Math., 2 (1959), 93-109. 22. Petrov V.V., Sums of idependent random variables, New York: Springer, 1975. 23. Billingsley P., Probability and measure, New York: Wiley, 1986.
Pages	122-137
Volume	49
Issue	2
Year	2018
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html