Classical Martingale Theory Research Articles

A remark on maximal functions for noncommutative martingales

Let \({\mathcal{M}}\) be a finite von Neumann algebra equipped with a normal tracial state τ. It is shown that if \({\{x_n\}_{n\geq1}}\) is a sequence of positive marginales that is bounded in \({L^1(\mathcal{M},\mathcal{T})}\), then for every 0 < p < 1, there exists \({y \in L^p(\mathcal{M},\mathcal{T})}\) satisfying the property that \({x_n \leq y}\) for all \({n\geq 1}\). Thus we obtain a noncommutative analogue of a maximal function theorem from classical martingale theory.

Marginal hazard regression for correlated failure time data with auxiliary covariates

In many biomedical studies, it is common that due to budget constraints, the primary covariate is only collected in a randomly selected subset from the full study cohort. Often, there is an inexpensive auxiliary covariate for the primary exposure variable that is readily available for all the cohort subjects. Valid statistical methods that make use of the auxiliary information to improve study efficiency need to be developed. To this end, we develop an estimated partial likelihood approach for correlated failure time data with auxiliary information. We assume a marginal hazard model with common baseline hazard function. The asymptotic properties for the proposed estimators are developed. The proof of the asymptotic results for the proposed estimators is nontrivial since the moments used in estimating equation are not martingale-based and the classical martingale theory is not sufficient. Instead, our proofs rely on modern empirical process theory. The proposed estimator is evaluated through simulation studies and is shown to have increased efficiency compared to existing methods. The proposed method is illustrated with a data set from the Framingham study.

Maximal inequalities for [formula omitted]-martingales

Maximal inequalities play an important role in the classical martingale theory. This paper studies maximal inequalities for g -martingales under the g -expectation framework. Two kinds of maximal inequalities for g -martingales are obtained.

Martingales in Banach Lattices

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (En) on a Banach lattice F is said to be a filtration if EnEm = En∧m. A sequence (xn )i nF is a martingale if Enxm = xn whenever n m. Denote by M = M(F,(E n)) the Banach space of all norm uniformly bounded martingales. It is shown that if F doesn't contain a copy of c0 or if every En is of finite rank then M is itself a Banach lattice. Convergence of martingales is inves- tigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings F� → M� → F ∗∗ . Finally, it is shown that every martingale difference sequence is a monotone basic sequence. In this paper, we define a martingale in terms of Banach lattices. Rephras- ing the title of (29), this paper could be entitled Martingales without prob- ability. We start with a brief review of classical martingale theory and Banach lattices.

Square function inequalities for non-commutative martingales

We prove a non-commutative version of the weak-type (1,1) boundedness of square functions of martingales. More precisely, we prove that there is an absolute constantK with the following property: ifM is a semifinite von Neumann algebra with a faithful normal traceτ and (M n ) n=1 ∞ is an increasing filtration of von Neumann subalgebras of (M then for any martingalex= n=1 ∞ inL 1(M,τ), adapted to (M n ) n=1 ∞ , there is a decomposition into two sequences (x n ) n=1 ∞ and (z n ) n=1 ∞ withx n=y n+z nfor everyn≥1 and such that\(\left\| {\left( {\sum\limits_{n = 1}^\infty {\left| {dy_n } \right|^2 } } \right)^{1/2} } \right\|_{1,\infty } + \left\| {\left( {\sum\limits_{n = 1}^\infty {\left| {dz_n^ * } \right|^2 } } \right)^{1/2} } \right\|_{1,\infty } \leqslant K\left\| x \right\|_1 \). This generalizes a result of Burkholder from classical martingale theory to non-commutative martingales. We also include some applications to martingale Hardy spaces.

Non-Commutative Martingale Transforms,

We prove that non-commutative martingale transforms are of weak type (1,1). More precisely, there is an absolute constant C such that if M is a semi-finite von Neumann algebra and (Mn)n=1∞ is an increasing filtration of von Neumann subalgebras of M then for any non-commutative martingale x=(xn)n=1∞ in L1(M), adapted to (Mn)n=1∞, and any sequence of signs (εn)n=1∞,∥ε1x1+∑n=2Nεn(xn−xn−1)∥1,∞⩽C∥xN∥1 for every N⩾2. This generalizes a result of Burkholder from classical martingale theory to non-commutative setting and answers positively a question of Pisier and Xu. As applications, we get the optimal order of the unconditional Martingale differences (UMD)-constants of the Schatten class Sp when p→∞. Similarly, we prove that the UMD-constant of the finite-dimensional Schatten class Sn1 is of order log(n+1). We also discuss the Pisier–Xu non-commutative Burkholder–Gundy inequalities.

Non-Commutative Martingale Transforms

We prove that non-commutative martingale transforms are of weak type (1,1). More precisely, there is an absolute constant C such that if M is a semi-finite von Neumann algebra and (Mn)n=1∞ is an increasing filtration of von Neumann subalgebras of M then for any non-commutative martingale x=(xn)n=1∞ in L1(M), adapted to (Mn)n=1∞, and any sequence of signs (εn)n=1∞,∥ε1x1+∑n=2Nεn(xn−xn−1)∥1,∞⩽C∥xN∥1 for every N⩾2. This generalizes a result of Burkholder from classical martingale theory to non-commutative setting and answers positively a question of Pisier and Xu. As applications, we get the optimal order of the unconditional Martingale differences (UMD)-constants of the Schatten class Sp when p→∞. Similarly, we prove that the UMD-constant of the finite-dimensional Schatten class Sn1 is of order log(n+1). We also discuss the Pisier–Xu non-commutative Burkholder–Gundy inequalities.