Convergence of weighted averages of martingales in Banach function spaces
Banach function spaces are Banach spaces of random variables between \(L^\infty\) and \(L^1\) with well-behaved norms. Examples are \(L^p\)-spaces, Orlicz spaces and Lorentz spaces. When \(f= (f_n)\) is a martingale and \((w_n)\) a sequence of positive numbers such that \(W_n= \sum^n_{k=1} w_k\to \infty\), \textit{M. Izumisawa} and \textit{N. Kazamaki} [Tôhoku Math. J., II. Ser. 29, 115-124 (1977; Zbl 0359.60050)] proved that \(f\) converges in \(L^p\) if and only if the weighted averages \(W^{-1}_n \sum^n_{k=1} w_kf_k\) converge in \(L^p\). Under an additional condition, this result is extended to most Banach function spaces. It also works when \(X\) is a weighted \(L^p\)-space with a weight function satisfying the usual \(A_p\) condition.
scientific article; zbMATH DE number 1441386