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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 2, Fasc. 1,
pages 83 - 88
 

ON BOUNDEDNESS AND CONVERGENCE OF SOME BANACH SPACE VALUED RANDOM SERIES

Rafał Sztencel

Abstract: Let (f ) i and (g) i be sequences of independent symmetric random variables and (x ) i a sequence of elements from a Banach space. We prove that under certain assumptions the a.s. boundedness of the scries sum xf ii implies the a.s. convergence of sum x g ii in every Banach space.

If f i are identically distributed, E |f | i is finite, g i are identically distributed and non-degenerate, then the above implication fails in c . 0

If f i are equidistributed and there is a sequence (a ) n such that

sum u a-R 1 |fi|--> 1 in probability, i=1
then there is a sequence (xi) in c0 such that sum xifi is a.s. bounded, but does not converge a.s.

In particular, if fi are p -stable with Eeitfn = e-|t|p, then for p < 1 the a.s. boundedness of the series implies its a.s. convergence, but for p > 1 it fails.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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