Geodesic
On probability and moment inequalities for supermartingales and martingales
Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 2, pp. 391-400

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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 $S_k$, $k=1,\dots,n$ is a supermartingale. This inequality is stated in terms of probabilities $\mathbf P(X_j>y)$ and conditional variances of random variables $X_j$, $j=1,\dots,n$. 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, supermartingale, Burkholder inequality, Bernstein and Bennet–Hoeffding inequalities, Rosenthal inequality, Fuk's inequality, separable Banach space, filtered probability space.
Mots-clés : martingale 
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     author = {S. V. Nagaev},
     title = {On probability and moment inequalities for supermartingales and martingales},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {391--400},
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     volume = {51},
     number = {2},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a9/}
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S. V. Nagaev. On probability and moment inequalities for supermartingales and martingales. Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 2, pp. 391-400. http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a9/