ON BOUNDEDNESS AND CONVERGENCE OF SOME BANACH SPACEVALUED RANDOM SERIES
Rafał Sztencel
Abstract: Let and be sequences of independent symmetric random variables and
a sequence of elements from a Banach space. We prove that under certain assumptions
the a.s. boundedness of the scries implies the a.s. convergence of in every
Banach space.
If are identically distributed, is finite, are identically distributed and
non-degenerate, then the above implication fails in
If are equidistributed and there is a sequence such that
then
there is a sequence in such that is a.s. bounded, but does not converge
a.s.
In particular, if are -stable with then for the a.s.
boundedness of the series implies its a.s. convergence, but for it fails.