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Teor. Veroyatnost. i Primenen., 2006, Volume 51, Issue 2, Pages 391–400 (Mi tvp62)  

This article is cited in 2 scientific papers (total in 2 papers)

Short Communications

On probability and moment inequalities for supermartingales and martingales

S. V. Nagaev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: The probability inequality for $\max_{k\le n}S_k$, where $S_k=\sum_{j=1}^kX_j$, is proved under the assumption that the sequence $S_k$, $k=1,…,n$ is a supermartingale. This inequality is stated in terms of probabilities $\mathbf P(X_j>y)$ and conditional variances of random variables $X_j$, $j=1,…,n$. As a simple consequence the well-known moment inequality due to Burkholder is deduced. Numerical bounds are given for constants in Burkholder's inequality.

Keywords: expectation, martingale, supermartingale, Burkholder inequality, Bernstein and Bennet–Hoeffding inequalities, Rosenthal inequality, Fuk's inequality, separable Banach space, filtered probability space.

DOI: https://doi.org/10.4213/tvp62

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English version:
Theory of Probability and its Applications, 2007, 51:2, 367–377

Bibliographic databases:

Received: 11.06.2002
Revised: 14.04.2005

Citation: S. V. Nagaev, “On probability and moment inequalities for supermartingales and martingales”, Teor. Veroyatnost. i Primenen., 51:2 (2006), 391–400; Theory Probab. Appl., 51:2 (2007), 367–377

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Nagaev S.V., “On probability and moment inequalities for supermartingales and martingales”, Acta Appl. Math., 97:1-3 (2007), 151–162  crossref  mathscinet  zmath  isi  elib  scopus
    2. E. L. Presman, “Estimation of the Constant in a Burkholder Inequality for Supermartingales and Martingales”, Theory Probab. Appl., 53:1 (2009), 173–179  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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