Some Inequalities for Tree Martingales

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 , t ∈ T) and E[σ ^(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.