Almost sure convergence for double arrays of block-wise M-dependent random elements in Banach spaces

作者：Nguyen Van Quang^*; Le Van Thanh; Nguyen Duy Tien

来源：Georgian Mathematical Journal, 2011, 18(4): 777-800.

DOI：10.1515/GMJ.2011.0045

• Blockwise M-dependent random elements
• strong law of large numbers
• double array of random elements
• Rademacher type p Banach space
• stable type p Banach space

For a double array of blockwise M-dependent random elements {Vmn : m >= 1, n >= 1} taking values in a real separable Rademacher type p (1 <= p <= 2) Banach space, we provide conditions to obtain the almost sure convergence for double sums Sigma(m)(i=1) Sigma(n)(j=1) Vij, m >= 1, n >= 1 . The paper treats two cases: (i) V-mn : m >= 1, n >= 1} is block-wise M-dependent with EVmn = 0, m, n >= 1, and (ii) V-mn : m >= 1, n >= 1} is block-wise p-orthogonal. The conditions for case (i) are shown to provide exact characterizations of Rademacher type p and stable type p Banach spaces. Examples are given showing that the conditions cannot be removed or weakened. It is also demonstrated that some of the well-known theorems in the literature are special cases of our results.

• 出版日期2011