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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 27, Fasc. 2,
pages 205 - 222
 

ON THE STRONG LAW OF LARGE NUMBERS FOR SEQUENCES OF BLOCKWISE INDEPENDENT AND BLOCKWISE $p$-ORTHOGONAL RANDOM ELEMENTS IN RADEMACHER TYPE $p$ BANACH SPACES

Andrew Rosalsky
Le Van Thanh

Abstract: For a sequence of random elements $\(V_n, n \geq 1\)$ taking values in a real separable Rademacher type $p$ ($1\leq p\leq 2$) Banach space and positive constants $b_n \uparrow \infty $, conditions are provided for the strong law of large numbers $\sum _(i=1)^n V_i/b_n \to 0$ almost surely. We treat the following cases: (i) $\(V_n, n \geq 1\)$ is blockwise independent with $EV_n = 0$, $n\geq 1$, and (ii) $\(V_n, n \geq 1\)$ is blockwise $p$-orthogonal. The conditions for case (i) are shown to provide an exact characterization of Rademacher type $p$ 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 $p$-orthogonal random elements, strong law of large numbers, almost sure convergence, Rademacher type $p$ Banach space.

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