
Estimates from below for characteristic functions of probability
laws 
Author 
M. I. Parolya, M. M. Sheremeta
marta0691@rambler.ru, m_m_sheremeta@list.ru
Ivan Franko Natonal University of Lviv

Abstract 
Let $\varphi$ be the characteristic function of a probability law F that is analytic in $\mathbb{D}_{R}=\{z: z<R \},$ $0<R\leq+\infty,$ $M(r,\varphi)=\max\left\{\varphi(z)\colon z=r<R\right\}$ and $W_{F}(x)=1F(x)+F(x),$ $x\geq 0.$ A connection between the growth of $M(r,\varphi)$ and the decrease of $W_{F}(x)$ is investigated in terms of estimates from below. For entire characteristic functions is proved, for example, that if $\ln x_k\geq \lambda\ln(\frac{1}{x_k}\ln\frac{1}{W_{F}(x_k)})$ for some increasing sequence $(x_k)$ such that $x_{k+1}=O(x_k), k\rightarrow\infty,$ then $\ln\frac{\ln M(r,\varphi)}{r}\geq (1+o(1))\lambda\ln r$ as $r\rightarrow+\infty.$ 
Keywords 
characteristic function; probability law; lower estimate 
Reference 
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Pages 
5466 
Volume 
39 
Issue 
1 
Year 
2013 
Journal 
Matematychni Studii 
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