Bernoulli

Inequalities for dominated martingales

Abstract

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.

Article information

Source
Bernoulli, Volume 13, Number 1 (2007), 54-79.

Dates
First available in Project Euclid: 30 March 2007

https://projecteuclid.org/euclid.bj/1175287720

Digital Object Identifier
doi:10.3150/07-BEJ5151

Mathematical Reviews number (MathSciNet)
MR2307394

Zentralblatt MATH identifier
1125.60037

Citation

Osękowski, Adam. Inequalities for dominated martingales. Bernoulli 13 (2007), no. 1, 54--79. doi:10.3150/07-BEJ5151. https://projecteuclid.org/euclid.bj/1175287720