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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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