대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제47권3호
  • /
  • Pages.467-482
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  • 2010
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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MEAN CONVERGENCE THEOREMS AND WEAK LAWS OF LARGE NUMBERS FOR DOUBLE ARRAYS OF RANDOM ELEMENTS IN BANACH SPACES

  • Dung, Le Van (Faculty of Mathematics Danang University of Education) ;
  • Tien, Nguyen Duy (Faculty of Mathematics National University of Hanoi)
  • 투고 : 2008.08.08
  • 발행 : 2010.05.31
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초록

For a double array of random elements {$V_{mn};m{\geq}1,\;n{\geq}1$} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which $k_{mn}^{-\frac{1}{r}}\sum{{u_m}\atop{i=1}}\sum{{u_n}\atop{i=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$ in $L_r$ (0 < r < 2). The weak law results provide conditions for $k_{mn}^{-\frac{1}{r}}\sum{{T_m}\atop{i=1}}\sum{{\tau}_n\atop{j=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$ in probability where {$T_m;m\;{\geq}1$} and {${\tau}_n;n\;{\geq}1$} are sequences of positive integer-valued random variables, {$k_{mn};m{{\geq}}1,\;n{\geq}1$} is an array of positive integers. The sharpness of the results is illustrated by examples.

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참고문헌

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피인용 문헌

  1. A new family of convex weakly compact valued random variables in Banach space and applications to laws of large numbers vol.82, pp.1, 2012, https://doi.org/10.1016/j.spl.2011.08.012