Estimates from below for characteristic functions of probability laws
Let $varphi$ be the characteristic function of a probabilitylaw F that is analytic in $mathbb{D}_{R}={zcolon |z|<R},$ $0<Rleq+infty,$ $M(r,varphi)=maxleft{|varphi(z)|colon|z|=r<Right}$ and $W_{F}(x)=1-F(x)+F(-x),$ $xgeq 0.$ Aconnection between the growth of $M(r,varphi)$ and thedecrease it...
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Ivan Franko National University of Lviv
2013-04-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/texts/2013/39_1/54-66.pdf |
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| author | M. I. Parolya M. M. Sheremeta |
| author_facet | M. I. Parolya M. M. Sheremeta |
| author_sort | M. I. Parolya |
| collection | DOAJ |
| description | Let $varphi$ be the characteristic function of a probabilitylaw F that is analytic in $mathbb{D}_{R}={zcolon |z|<R},$ $0<Rleq+infty,$ $M(r,varphi)=maxleft{|varphi(z)|colon|z|=r<Right}$ 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 $rightarrow+infty.$ |
| format | Article |
| id | doaj-art-16fbe6acf2db4b0a8a95c1483ba96b10 |
| institution | Kabale University |
| issn | 1027-4634 |
| language | deu |
| publishDate | 2013-04-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-16fbe6acf2db4b0a8a95c1483ba96b102025-08-20T03:33:27ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342013-04-013915466Estimates from below for characteristic functions of probability lawsM. I. ParolyaM. M. SheremetaLet $varphi$ be the characteristic function of a probabilitylaw F that is analytic in $mathbb{D}_{R}={zcolon |z|<R},$ $0<Rleq+infty,$ $M(r,varphi)=maxleft{|varphi(z)|colon|z|=r<Right}$ 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 $rightarrow+infty.$http://matstud.org.ua/texts/2013/39_1/54-66.pdfcharacteristic functionprobability lawlower estimate |
| spellingShingle | M. I. Parolya M. M. Sheremeta Estimates from below for characteristic functions of probability laws Математичні Студії characteristic function probability law lower estimate |
| title | Estimates from below for characteristic functions of probability laws |
| title_full | Estimates from below for characteristic functions of probability laws |
| title_fullStr | Estimates from below for characteristic functions of probability laws |
| title_full_unstemmed | Estimates from below for characteristic functions of probability laws |
| title_short | Estimates from below for characteristic functions of probability laws |
| title_sort | estimates from below for characteristic functions of probability laws |
| topic | characteristic function probability law lower estimate |
| url | http://matstud.org.ua/texts/2013/39_1/54-66.pdf |
| work_keys_str_mv | AT miparolya estimatesfrombelowforcharacteristicfunctionsofprobabilitylaws AT mmsheremeta estimatesfrombelowforcharacteristicfunctionsofprobabilitylaws |