Skip to main content Accessibility help
×
Home

two examples concerning martingales in banach spaces

Published online by Cambridge University Press:  04 October 2005

jörg wenzel
Affiliation:
department of mathematics and applied mathematics, university of pretoria, pretoria 0002, south africawenzel@minet.uni-jena.de
Get access

Abstract

the analytic concepts of martingale type $p$ and cotype $q$ of a banach space have an intimate relation with the geometric concepts of $p$-concavity and $q$-convexity of the space under consideration, as shown by pisier. in particular, for a banach space $x$, having martingale type $p$ for some $p{>}1$ implies that $x$ has martingale cotype $q$ for some $q{<}\infty$.

the generalisation of these concepts to linear operators was studied by the author, and it turns out that the duality above only holds in a weaker form. an example is constructed showing that this duality result is best possible.

so-called random martingale unconditionality estimates, introduced by garling as a decoupling of the unconditional martingale differences (umd) inequality, are also examined.

it is shown that the random martingale unconditionality constant of $l_\infty^{2^n}$ for martingales of length $n$ asymptotically behaves like $n$. this improves previous estimates by geiss, who needed martingales of length $2^n$ to show this asymptotic. at the same time the order in the paper is the best that can be expected.

Type
notes and papers
Copyright
the london mathematical society 2005

Access options

Get access to the full version of this content by using one of the access options below.

Footnotes

this paper grew out of the author's habilitation thesis, which was supported by dfg grant we 1868/1-1.

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 2 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 27th January 2021. This data will be updated every 24 hours.

Hostname: page-component-898fc554b-p5tlp Total loading time: 0.38 Render date: 2021-01-27T04:59:14.386Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

two examples concerning martingales in banach spaces
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

two examples concerning martingales in banach spaces
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

two examples concerning martingales in banach spaces
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *