RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor
Subscription
Guidelines for authors
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Teor. Veroyatnost. i Primenen.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 1, Pages 162–172 (Mi tvp165)  

This article is cited in 1 scientific paper (total in 1 paper)

Short Communications

Maximal $l\phi$-inequalities for nonnegative submartingales

U. Röslera, G. Alsmeyerb

a Christian-Albrechts-Universität
b Westfälische Wilhelms-Universität Münster

Abstract: Let $(M_n)_{n\ge 0}$ be a nonnegative submartingale and let $M_n^*\stackrel{\textrm{def}}{=}\max_{0\le k\le n}M_k$, $n\ge 0$, be the associated maximal sequence. For nondecreasing convex functions $\phi\colon[0,\infty)\to[0,\infty)$ with $\phi(0)=0$ (Orlicz functions) we provide various inequalities for $E\phi(M_n^*)$ in terms of $E\Phi_a(M_n)$, where, for $a\ge 0$,
$$ \Phi_{a}(x) \stackrel{\textrm{def}}{=} \int_{a}^{x}\int_{a}^{s}\frac{\phi'(r)}{r} dr ds, \qquad x>0. $$
Of particular interest is the case $\phi(x)=x$ for which a variational argument leads us to
$$ EM_n^*\le(1+(E(\int_{1}^{M_n\vee 1}\log x dx))^{1/2})^2. $$
A further discussion shows that the given bound is better than Doob's classical bound $e(e-1)^{-1}(1+\textbf E M_n\log^{+}M_n)$ whenever $\textbf E(M_n-1)^{+}\ge e-2\approx 0.718$.

Keywords: nonnegative submartingale, maximal sequence, Orlicz function, Young function, Choquet representation, convex function inequality.

DOI: https://doi.org/10.4213/tvp165

Full text: PDF file (888 kB)
References: PDF file   HTML file

English version:
Theory of Probability and its Applications, 2006, 50:1, 118–128

Bibliographic databases:

Received: 10.12.2003
Language:

Citation: U. Rösler, G. Alsmeyer, “Maximal $l\phi$-inequalities for nonnegative submartingales”, Teor. Veroyatnost. i Primenen., 50:1 (2005), 162–172; Theory Probab. Appl., 50:1 (2006), 118–128

Citation in format AMSBIB
\Bibitem{RosAls05}
\by U.~R\"osler, G.~Alsmeyer
\paper Maximal $l\phi$-inequalities for nonnegative submartingales
\jour Teor. Veroyatnost. i Primenen.
\yr 2005
\vol 50
\issue 1
\pages 162--172
\mathnet{http://mi.mathnet.ru/tvp165}
\crossref{https://doi.org/10.4213/tvp165}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2222745}
\zmath{https://zbmath.org/?q=an:1099.60032}
\elib{http://elibrary.ru/item.asp?id=9153113}
\transl
\jour Theory Probab. Appl.
\yr 2006
\vol 50
\issue 1
\pages 118--128
\crossref{https://doi.org/10.1137/S0040585X97981548}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000236850700009}


Linking options:
  • http://mi.mathnet.ru/eng/tvp165
  • https://doi.org/10.4213/tvp165
  • http://mi.mathnet.ru/eng/tvp/v50/i1/p162

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Wang X., Hu Sh., “On the Maximal Inequalities For Conditional Demimartingales”, J. Math. Inequal., 8:3 (2014), 545–558  crossref  mathscinet  zmath  adsnasa  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
    Number of views:
    This page:169
    Full text:54
    References:47

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020