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Some Inequalities for Tree Martingales
Authors:Tong-jun He  You-liang Hou
Institution:(1) College of Mathematics and Computer Science, Fuzhou University, Fuzhou 35002, China;(2) School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Abstract:Abstract   In this paper we study tree martingales and proved that if 1 ≤ α, β < ∞, 1 ≤ p < ∞ then for every predictable tree martingale f = (f t , tT) and Eσ (P)(f)] < ∞, E S (P)(f)] < ∞, it holds that
$$
\begin{aligned}
  & \left\| {{\left( {S^{{{\left( p \right)}}}_{t} {\left( f \right)},t \in T} \right)}\left\| {_{{M^{{\alpha \infty }} }} } \right.} \right. \leqslant C_{{\alpha \beta }} \left\| f \right.\left\| {_{{p^{{\alpha \beta }} }} } \right., \\
 & \left\| {{\left( {\sigma ^{{{\left( p \right)}}}_{t} {\left( f \right)},t \in T} \right)}} \right.\left\| {_{{M^{{\alpha \infty }} }} } \right. \leqslant C_{{\alpha \beta }} \left\| f \right.\left\| {_{{p^{{\alpha \beta }} }} } \right., \\ 
 \end{aligned} 
$$
where C αβ depends only on α and β. Supported by The National Natural Science Foundation of China (No.10371093)
Keywords:Tree martingale  quadratic variation  conditional quadratic variation
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