English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

If      f = ( f 1 , f 2 , …  ) f = ({f_1},{f_2}, \ldots )       is a real      L 1 {L^1}      -bounded martingale and      ε  &gt; 0 \varepsilon &gt; 0      , then there is a martingale  g  of bounded variation satisfying      ‖ f −  g ‖ 1 &gt; ε  {\left \| {f - g} \right \|_1} &gt; \varepsilon      . The same result holds for  X -valued martingales, where  X  is a Banach space, provided  X  has the Radon-Nikodým property. In fact, this characterizes Banach spaces having the Radon-Nikodým property. Theorem 1 identifies, for an arbitrary Banach space, the class of      L 1 {L^1}      -bounded martingales that converge almost everywhere.

Approximation of 𝐿¹-bounded martingales by martingales of bounded variation