Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 2, 330--343

On the Weak Law with Random Indices for Arrays of Banach Space Valued Random Elements

Andrew Rosalsky, University of Florida, USA
Andrei Volodin, University of Regina, Canada

SUMMARY. For a sequence of constants $\{a_n, \ n\ge 1\}$, an array of rowwise independent and stochastically dominated random elements $\{V_{nj}, \ j\ge 1, n\ge 1\}$ in a real separable Rademacher type $p$ Banach space for some $p \in [1,2]$, and a sequence of positive integer-valued random variables $\{T_n, \ n\ge 1\}$, a general weak law of large numbers of the form $\sum_{j=1}^{T_n} a_j (V_{nj} - c_{nj})/ b_{\lfloor \alpha_n\rfloor} \stackrel{P}{\longrightarrow} 0$ is established, where $\{c_{nj}, \ j \geq 1, n \geq 1\}$ is an array of truncated expectations, and $\alpha_n \rightarrow \infty, b_n \rightarrow \infty$ are suitable sequences. No assumption is made concerning the existence of expected values or absolute moments of the random elements $\{V_{nj}, \ j \geq 1, n \geq 1\}$. The current work is a new version of a result of Adler, Rosalsky, and Volodin ({\it J. Theoret. Probab.} vol.\,10, 1997, 605--623).

AMS (2000) subject classification. Primary 60B12; secondary 60B11.

Key words and phrases. Rademacher type $p$ Banach space, array of rowwise independent random elements, weighted sums, weak law of large numbers, random indices.

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