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

✍ Scribed by Van Quang, Nguyen (author);Van Thanh, Le (author);Tien, Nguyen Duy (author)

• Book ID: 120601951
• Publisher: Walter de Gruyter GmbH & Co. KG
• Year: 2011
• Tongue: English
• Weight: 291 KB
• Volume: 18
• Category: Article
• ISSN: 1072-947X
• DOI: 10.1515/GMJ.2011.0045

✦ Synopsis

For a double array of blockwise M-dependent random elements ¹V mn W m 1; n 1º taking values in a real separable Rademacher type p (1 Ä p Ä 2) Banach space, we provide conditions to obtain the almost sure convergence for double sums

The paper treats two cases: (i) ¹V mn W m 1; n 1º is block-wise M-dependent with EV mn D 0, m; n 1, and (ii) ¹V mn W m 1; n 1º 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.