ON THE STRONG LAW OF LARGE NUMBERS FOR SEQUENCES OF
BLOCKWISE INDEPENDENT AND BLOCKWISE
-ORTHOGONAL
RANDOM ELEMENTS IN RADEMACHER TYPE
BANACH SPACES
Andrew Rosalsky
Le Van Thanh
Abstract: For a sequence of random elements
taking values in a real separable Rademacher type
() Banach space and
positive constants ,
conditions are provided for the strong law of large numbers
almost surely. We treat
the following cases: (i) is
blockwise independent with ,
, and (ii)
is blockwise
-orthogonal.
The conditions for case (i) are shown to provide an exact characterization of Rademacher
type
Banach spaces. The current work extends results of Móricz [12], Móricz et al. [13], and
Gaposhkin [8]. Special cases of the main results are presented as corollaries and illustrative
examples or counterexamples are provided.
2000 AMS Mathematics Subject Classification: Primary 60F15; Secondary: 60B11,
60B12.
Key words and phrases: Blockwise independent random elements, blockwise
-orthogonal
random elements, strong law of large numbers, almost sure convergence, Rademacher type
Banach space.