Properties of local orthonormal systems Part I: Unconditionality in Lp$L^p$, 1

Jacek Gulgowski, Anna Kamont and Markus Passenbrunner

Mathematische Nachrichten, 2024, vol. 297, issue 5, 1838-1865

Abstract: Assume that we are given a filtration (Fn)$(\mathcal F_n)$ on a probability space (Ω,F,P)$(\Omega,\mathcal F,\mathbb {P})$ of the form that each Fn$\mathcal F_n$ is generated by the partition of one atom of Fn−1$\mathcal F_{n-1}$ into two atoms of Fn$\mathcal F_n$ having positive measure. Additionally, assume that we are given a finite‐dimensional linear space S$S$ of F$\mathcal F$‐measurable, bounded functions on Ω$\Omega$ so that on each atom A$A$ of any σ$\sigma$‐algebra Fn$\mathcal F_n$, all Lp$L^p$‐norms of functions in S$S$ are comparable independently of n$n$ or A$A$. Denote by Sn$S_n$ the space of functions that are given locally, on atoms of Fn$\mathcal F_n$, by functions in S$S$ and by Pn$P_n$ the orthoprojector (with respect to the inner product in L2(Ω)$L^2(\Omega)$) onto Sn$S_n$. Since S=span{1Ω}$S = \operatorname{span}\lbrace \mathbbm 1_\Omega \rbrace$ satisfies the above assumption and Pn$P_n$ is then the conditional expectation En$\mathbb {E}_n$ with respect to Fn$\mathcal F_n$, for such filtrations, martingales (Enf)$(\mathbb {E}_n f)$ are special cases of our setting. We show in this article that certain convergence results that are known for martingales (or rather martingale differences) are also true in the general framework described above. More precisely, we show that the differences (Pn−Pn−1)f$(P_n - P_{n-1})f$ form an unconditionally convergent series and are democratic in Lp$L^p$ for 1

Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.202300225

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:297:y:2024:i:5:p:1838-1865

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().