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February 2007 Inequalities for dominated martingales
Adam Osękowski
Bernoulli 13(1): 54-79 (February 2007). DOI: 10.3150/07-BEJ5151


Let $(M_n)$, $(N_n)$ be two Hilbert-space-valued martingales adapted to some filtration $(ℱ_n)$, with corresponding difference sequences $(d_n)$, $(e_n)$, respectively. We assume that $(N_n)$ weakly dominates $(M_n)$, that is, for any convex non-decreasing function $ϕ : ℝ_+→ℝ-_+$ and any $n=1,2,…$ we have, almost surely, $\mathrm{E}(ϕ(|d_n|)|ℱ_{n−1})\leqslant\mathrm{E}(ϕ(|e_n|)|ℱ_{n−1})$. We apply the Burkholder method to show that for a convex non-decreasing function $\mathbf{Φ} : ℝ_+→ℝ_+$ satisfying some extra conditions we have, for any $n=1,2,…$, $‖M_n‖_\mathbf{Φ}≤C_\mathbf{Φ}‖N_n‖_\mathbf{Φ}$, where $‖⋅‖_\mathbf{Φ}$ denotes an Orlicz norm with respect to $\mathbf{Φ}$ and $C_\mathbf{Φ}$ is a constant which depends only on $\mathbf{Φ}$. This approach unifies and extends the classical Burkholder inequalities for subordinated martingales and the inequalities for tangent martingales. The method leads to moment inequalities for Rosenthal-type dominated martingales and variance-dominated Gaussian martingales. All the constants obtained in the moment inequalities are of optimal order.


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Adam Osękowski. "Inequalities for dominated martingales." Bernoulli 13 (1) 54 - 79, February 2007.


Published: February 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1125.60037
MathSciNet: MR2307394
Digital Object Identifier: 10.3150/07-BEJ5151

Keywords: Martingales , Orlicz space , subordinated martingales

Rights: Copyright © 2007 Bernoulli Society for Mathematical Statistics and Probability


Vol.13 • No. 1 • February 2007
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