A mean convergence theorem and weak law for arrays of random elements in martingale type p Banach spaces

✍ Scribed by André Adler; Andrew Rosalsky; Andrej I. Volodin

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 393 KB
• Volume: 32
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(97)85593-9

✦ Synopsis

For weighted sums of the form Sn = Ejknl anj (Vnj--Cnj) where {anj, 1 <<.j<~kn < oo, n~> 1} are constants, {V~j, 1 <~j<~k~, n>~l} are random elements in a real separable martingale type p Banach space, and {cnj, 1 <<.j<~kn, n>>-1} are suitable conditional expectations, a mean convergence theorem and a general weak law of large numbers are established. These results take the form Ilsnll ~r 0 and S~ ~ 0, respectively. No conditions are imposed on the joint distributions of the {Vnj, 1 <<.j~k., n>~l}. The mean convergence theorem is proved assuming that {ll~jHr, l<~j<~k~, n>~l} is (la.jlr}uniformly integrable whereas the weak law is proved under a Ceshro type condition which is weaker than Ces~ro uniform integrability. The sharpness of the results is illustrated by an example. The current work extends that of Gut (1992) and Hong and Oh (1995).