Asymmetric Doob inequalities in continuous time

In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let $(\M,\tau)$ be a noncommutative probability space equipped with a weak-$*$ dense filtration of von Neumann subalgebras $(\M_n)_{n \ge 1}$. Let $\E_n$ denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for $1 < p < 2$ and $x \in L_p(\M,\tau)$ one can find $a, b \in L_p(\M,\tau)$ and contractions $u_n, v_n \in \M$ such that $$\E_n(x) = a u_n + v_n b \quad \mbox{and} \quad \max \big\{ \|a\|_p, \|b\|_p \big\} \le c_p \|x\|_p.$$ Moreover, it turns out that $a u_n$ and $v_n b$ converge in the row/column Hardy spaces $\H_p^r(\M)$ and $\H_p^c(\M)$ respectively. In particular, this solves a problem posed by Defant and Junge in 2004. In the case $p=1$, our results establish a noncommutative form of Davis celebrated theorem on the relation between martingale maximal and square functions in $L_1$, w...

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

Let 1 ≤ p < ∞ and (x n) n∈N be a sequence of positive elements in a noncommutative L p space and (E n) n∈N be an increasing sequence of conditional expectations, then n E n (x n) p ≤ c p n x n p. This inequality is due to Burkholder, Davis and Gundy in the commutative case. By duality, we obtain a version of Doob's maximal inequality for 1 < p ≤ ∞.

Stochastics, 2021

We employ some techniques involving projections in a von Neumann algebra to establish some maximal inequalities such as the strong and weak symmetrization, Lévy, Lévy-Skorohod, and Ottaviani inequalities in the realm of quantum probability spaces.

Bulletin of the London Mathematical Society, 2005

We determine the optimal orders for the best constants in the non-commutative Burkholder-Gundy, Doob and Stein inequalities obtained recently in the non-commutative martingale theory.

2013

We introduce Hardy spaces for martingales with respect to continuous filtration for von Neumann algebras. In particular we prove the analogues of the Burkholder/Gundy and Burkholder/Rosenthal inequalities in this setting. The usual arguments using stopping times in the commutative case are replaced by tools from noncommutative function theory and allow us to obtain the analogue of the Feffermann-Stein duality and prove a noncommutative Davis decomposition.

Illinois Journal of Mathematics

We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type inequality. We also provide a noncommutative Azuma inequality for noncommutative supermartingales in which instead of a fixed upper bound for the variance we assume that the variance is bounded above by a linear function of variables. We then employ it to deduce a noncommutative Bernstein inequality and an inequality involving L p-norm of the sum of a martingale difference.

Acta Mathematica Scientia, 2017

In this paper we prove the existence of conditional expectations in the noncommutative L p (M, Φ) spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the norm convergence of the martingales in noncommutative L p (M, Φ) spaces.

Probability Theory and Related Fields, 2014

We prove a deviation inequality for noncommutative martingales by extending Oliveira's argument for random matrices. By integration we obtain a Burkholder type inequality with satisfactory constant. Using continuous time, we establish noncommutative Poincaré type inequalities for "nice" semigroups with a positive curvature condition. These results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to Bobkov and Götze in the commutative case. To demonstrate our setting is general enough, we give various examples, including certain group von Neumann algebras, random matrices and classical diffusion processes, among others.

Journal of the American Mathematical Society

This paper is devoted to the study of various maximal ergodic theorems in noncommutative L p L_p -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on L p L_p and the analogue of Stein’s maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

The Michigan Mathematical Journal

Based on a maximal inequality type result of Cuculescu, we establish some noncommutative maximal inequalities such as Hajék-Penyi inequality and Etemadi inequality. In addition, we present a noncommutative Kolmogorov type inequality by showing that if x 1 , x 2 ,. .. , x n are successively independent self-adjoint random variables in a noncommutative probability space (M, τ) such that τ (x k) = 0 and s k s k−1 = s k−1 s k , where s k = k j=1 x j , then for any λ > 0 there exists a projection e such that 1 − (λ + max 1≤k≤n x k) 2 n k=1 var(x k) ≤ τ (e) ≤ τ (s 2 n) λ 2. As a result, we investigate the relation between convergence of a series of independent random variables and the corresponding series of their variances.