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Some limit theorems for weighted sums of random variable fields

Authors:	Gan Shixin Chen Pingyan

Institution:	(1) School of Mathematics and Statistics, Wuhan University, 430072 Wuhan Hubei, China;(2) Department of Mathematics, Jinan University, 510630 Guangzhou Guangdong, China

Abstract:	Let $\{ X_{\overline n } ,\overline n \in N^d \}$ be a field of Banach space valued random variables, O<r<p≤2 and $\{ a_{\overline n ,\overline k } ,(\overline n ,\overline k \in N^d \times N^d ,\overline k \leqslant \overline n \}$ a triangular array of real numbers, whereN ^d is thed-dimensional lattice (d≥1). Under the minimal condition that $\{ \|\|X_{\overline n } \|\|^r ,\overline n \in N^d \}$ is $\{ \|a_{\overline n ,\overline k } \|^r ,(\overline n ,\overline k )\} \in N^d \times N^d ,\overline k \leqslant \overline n \}$ integrable, we show that $\sum\limits_{_{\overline k \leqslant \overline n } } {a_{\overline n ,\overline k } } X_{\overline k } \underrightarrow {{\text{(or a}}{\text{.s)}}}$ 0 as $\|\overline n \| \to \infty$ . In the above if 0<r<1, the random variables are not needed to be independent. If 1≤r<p≤2, and Banach space valued random variables are independent with mean zero we assume the Banach space is of typep. If 1≤r<p≤2 and Banach space valued random variables are not independent we assume the Banach space isp-smoothable. Foundation item: Supported by the National Natural Science Foundation of China(10071058) Biography: GAN Shixin(1939-), male, Professor, research direction martingale theory, probability limiting theory and Banach space geometry theory.

Keywords:	Banach space of type p multidimensional index strong law of large numbers Lr convergence weighted sums of random variable fields martingale difference array
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