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Cím:A. S. behavior of the Ces\`aro means for quasi-orthogonal random fields.
Szerző:M\'oricz, F.
Forrás:Limit theorems in probability and statistics, Proc. 3rd Colloq., P\'ecs/Hung. 1989, Colloq. Math. Soc. J\'anos Bolyai 57, 425-441 (1990).
Nyelv:English
Absztrakt:[For the entire collection see Zbl 0713.00013.] \par Let $\{X\sb{ik}:$ $i,k=0,1,...\}$ be a random field such that $(i)\quad E X\sp 2\sb{ik}=\sigma\sp 2\sb{ik}<\infty,\quad (ii)\quad \vert E X\sb{ik}X\sb{j\ell}\vert \le \rho (\vert i-j\vert,\vert k-\ell \vert)\sigma\sb{ik}\sigma\sb{j\ell},(iii)\quad \sum\sp{\infty}\sb{m=0}\sum\sp{\infty}\sb{n=0}\rho (m,n)<\infty.$For $\alpha,\beta >-1$ we define the Ces\`aro means by $$ \zeta\sp{\alpha \beta}\sb{mn}:=(A\sp{\alpha}\sb mA\sp{\beta}\sb n)\sp{- 1}\sum\sp{m}\sb{i=0}\sum\sp{n}\sb{k=0}A\sp{\alpha -1}\sb{m-i}A\sp{\beta - 1}\sb{n-k}X\sb{ik}, $$ where $A\sp{\alpha}\sb 0:=1$ and $A\sp{\alpha}\sb m:=(\alpha +1)(\alpha +2)...(\alpha +m)/m!$. In {\S} 2 we provide sufficient conditions for a.s. convergence of $\zeta\sp{\alpha \beta}\sb{mn}$ to zero depending on the values of $\alpha$, $\beta$. In {\S} 3 we show that these conditions are the best possible or even necessary in certain special cases. In {\S} 6 we deal briefly with Banach space valued random variables. We raise three open problems and two possible extensions of our study.} \RV{F.M\'oricz (Szeged) 
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