Estimates from below for characteristic functions of probability laws - M. I. Parolya; M. M. Sheremeta (Математичні Студії)
Let $varphi$ be the characteristic function of a probabilitylaw F that is analytic in $mathbbD_{R}={zcolon |z|<R},$ $0<Rleq+infty,$ $M(r,varphi)=maxleft{|varphi(z)|colon|z|=r<Right}$ and $W_{F}(x)=1-F(x)+F(-x),$ $xgeq 0.$ Aconnection between the growth of $M(r,varphi)$ and thedecrease it of $W_{F}(x)$ is investigated in terms of estimatesfrom below. For entire characteristic functions {it} is proved,for example, that if $ln x_kgeqlambdaln(frac{1}{x_k}lnfrac{1}{W_{F}(x_k)})$ for someincreasing sequence $(x_k)$ such that $x_{k+1}=O(x_k),kightarrowinfty,$ then $lnfrac{ln M(r,varphi)}{r}geq(1+o(1))lambdaln r$ as $rightarrow+infty.$