Estimates from below for characteristic functions of probability laws

Author M. I. Parolya, M. M. Sheremeta,
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 $W_{F}(x)=1-F(x)+F(-x),$ $x\geq 0.$ 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
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Pages 54-66
Volume 39
Issue 1
Year 2013
Journal Matematychni Studii
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