Complete convergence of martingale arrays (Q1266766)

A sequence \(\{U_n, n\geq 1\}\) of random variables is said to converge completely to the constant \(c\) if \(\sum^\infty_{n= 1} P(| U_n- c|> \varepsilon)< \infty\) for all \(\varepsilon> 0\). The definition was introduced by \textit{P. L. Hsu} and \textit{H. Robbins} [Proc. Natl. Acad. Sci. USA 33, 25-31 (1947; Zbl 0030.20101)], and the authors also proved a result on complete convergence in the law of large numbers. The present paper is an extension to complete convergence for square-integrable martingale arrays together with several examples.