Estimates for the maximum modulus of analytic characteristic functions of probability laws on some sequences |
|
| Author |
marta0691@rambler.ru, m_m_sheremeta@list.ru
Ivan Franko National University of Lviv
|
| Abstract |
Let $\varphi$ be the characteristic function of a probability law $F$ analytic in $\mathbb{D}_{R}$,
$M(r,\varphi)=\max\{|\varphi(z)|\colon |z|=r\}$ and $W_F(x)=1-F(x)+F(-x)$, $x\geq 0$. We obtain upper estimates for $\mathop{\underline{\lim}}_{r\uparrow R}(\ln M(r,\varphi))/\Phi(r)$ for some positive convex on $(0,R)$ function $\Phi$ under certain
conditions on $W_F$.
|
| Keywords |
characteristic function; probability law; lower estimate
|
| Reference |
1. Linnik Ju.V., Ostrovskii I.V. Decomposition of random variables and vectors. – Moscow: Nauka, 1972,
479 p. (in Russian)
2. Skaskiv O.B., Sorokivs’kyi V.M. On the maximum modulus of characteristic functions of probability laws// Kraj. Zadachi Dyfer. Rivnyan’. – 2001. – V.7. – P. 286–289. (in Ukrainian) 3. Ramachandran B. On the order and the type of entire characteristic functions// Ann. Math. – 1962. – V.33, ¹4. – P. 1238–1255. 4. Jakovleva N.I. On the growth of entire characteristic functions of probability laws// Teoriya funkcii, func. analysis i ih prilozh. – 1971. – V.15. – P. 43–49. (in Russian) 5. Jakovleva N.I. On the growth of entire characteristic functions of probability laws// Problems of matematical physics and func. analysis, K.: Naukova Dumka, 1976. – P. 43–54. (in Russian) 6. Vynnyts’kyi B.V. On a property of entire characteristic functions of probability laws// Izv. Vuzov, Matem. – 1975. – V.4. – P. 95–97. (in Russian) 7. Dewess M. The tail behaviour of a distribution function and its connection to the growth of entire characteristic function// Math. Nachr. – 1978. – V.81. – P. 217–231. 8. Sorokivs’kyi V.M. On the growth of the analytic characteristic functions of probability laws// Izv. Vuzov, Matem. – 1979. – V.2. – P. 48–52. (in Russian) 9. Kinash O.M., Parolya M.I., Sheremeta M.M. Growth of the characteristic functions of probability laws// Dopovidi NAN Ukraine. – 2012. – V.8. – P. 13–17. (in Ukrainian) 10. Parolya M.I., Sheremeta M.M. Estimates from below for characteristic functions of probability laws// Mat. Stud. – 2013. – V.39, ¹1. – P. 54–66. 11. Sheremeta M.M., Sumyk O.M. A connection between the growth of Young conjugated functions// Mat. Stud. – 1999. – V.11, ¹1. – P. 41–47. (in Ukrainian) 12. Filevych P.V., Sheremeta M.M. On a theorem of L. Sons and assymptotical behaviour of Dirichlet series// Ukr. Math. Bull. – 2006. - V.3, ¹2. – P. 187–208. (in Ukrainian) 13. Zabolots’kyi M.V., Sheremeta M.M. Generalization of Lindeloff’s theorem// Ukr. Mat. Zh. – 1998. – V.50, ¹1. – P. 1117–1192. (in Ukrainian) 14. Sheremeta M.M. On two classes of positive functions and belonging to them of main characteristic of entire functions// Mat. Stud. – 2003. - V.19, ¹1. – P. 73–82. 15. Sumyk O.M., Sheremeta M.M. Estimates from below for maximal term of Dirichlet series// Izv. Vuzov, Matem. – 2001. – V.4. – P. 53–57. (in Russian) |
| Pages |
149-159
|
| Volume |
42
|
| Issue |
2
|
| Year |
2014
|
| Journal |
Matematychni Studii
|
| Full text of paper | |
| Table of content of issue |