Unbounded Order Convergence and Application to Martingales without Probability
By Niushan Gao, Foivos Xanthos. 2013
117 citations recorded by the source corpus.
A net $(x_\alpha)_{\alpha\in \Gamma}$ in a vector lattice $X$ is unbounded order convergent (uo-convergent) to $x$ if $|x_\alpha-x| \wedge y \xrightarrow{o} 0$ for each $y \in X_+$, and is unbounded order Cauchy (uo-Cauchy) if the net $(x_\alpha-x_{\alpha'})_{\Gamma\times \Gamma}$ is uo-convergent to 0. In the first part of this article, we study uo-convergent and uo-Cauchy nets in Banach lattices and use them to characterize Banach lattices with the positive Schur property and KB-spaces. In the second part, we use the concept of uo-Cauchy sequences to extend Doob's submartingale convergence theorems to a measure-free setting. Our results imply, in particular, that every norm bounded submartingale in $L_1(\Omega;F)$ is almost surely uo-Cauchy in $F$, where $F$ is an order continuous Banach lattice with a weak unit.
Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/417f49ec5d588f95c7335824eb9b4fa0715f5cf9
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Unbounded norm convergence in Banach lattices (2016, result): “0 in X. Indeed, let u = 0 ⊕ 1 = ∞ k=1gk. Then xn∧u = gn for every n, hence (xn∧u) does not converge to zero in X. The following three results are similar to Lemmas 3.6 and 3.7, and Proposition 3.9 of [GX14]; we replace uo-convergence with un-convergence, and we do not require the space be order cont”
- Uo-convergence and its applications to Cesàro means in Banach lattices (2015, methodology): “In [26], uo-convergence was applied in a study of abstract martingales in the framework of vector lattices.”