Probability and Mathematical Statistics: Vol. 10, Fasc. 1


	UNIVERSITY OF WROCŁAW

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	WROCŁAW UNIVERSITY OF SCIENCE AND TECHNOLOGY

Contents of PMS, Vol. 10, Fasc. 1, pages 161 - 168

TOPOLOGY OF THE CONVERGENCE IN PROBABILITY ON A LINEAR SPAN OF A SEQUENCE OF INDEPENDENT RANDOM VARIABLES K. Pietruska-Pałuba W. Smoleński Abstract: Let $X1,X2,...$ be a sequence of independent symmetric Hilbert space valued non-degenerated random variables and let $Lx$ denote the closed linear span of $(Xn)$ in $L0(_O_,F, P;H).$ If $Lx$ is a locally convex subspace of $L0,$ then $Lx$ is Banach iff $Lx$ does not contain an isomorphic copy of $R oo$ iff $sup P(Xn = 0) < 1. n$ If, moreover, $Xn$ are equidistributed and $P (Xn = 0) = 0,$ then $( ( ) ) -1- -1- Y (- Lx : P \|\|Y\|\|> 201 < 210$ is a bounded neighbourhood of zero. 2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -; Key words and phrases: -

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