ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 26, 28, 30, 46, 60 Shintani, Toshitada, Tomakomai National College of Technology, Japan Uniform integrability of L^{1}-bounded matingales. Let (X, | |) be Bnach space having Radon-Nikodým property. Consider Xvalued martingales on a probability space (Ω, a, P) and let L_{X}^{p}(Ω, a, P) (1 ≤ p ≤ ∞) be the set of all p the power Bochner integrable X-valued functions on (Ω, a, P). Then Theorem 1. Let f = (f_{n}) be a martingale. Then f is L_{X}^{1}-boundend ⇔ f is L_{X}^{p}-bounded (∀p > 1). Thus, L_{X}^{1}-bounded martingale f is uniformly integrable. Theorem 2. As set L_{X}^{1}(Ω, a, P) = L_{X}^{∞}(Ω, a, P). Theorem 3. L_{1}-bounded submartingale f = (f_{n}) is L^{∞}- bounded. Thus, f is uniformly integrable. Consider Hardy space on unit disc H^{p} ≡ H^{p}([−π, π], a, P) (⊂ L^{1}([−π, π], a, P)) (1 ≤ p ≤ ∞) with ∥ F ∥_{H^{∞}}≡ p → ∞_{lim} ∥ F ∥_{H^{p}}. Then Theorem 4. H^{1} = H^{∞}. Reference: T. Shintani, Martingale transfoms in a Banach space, (Submitted), (1997).