Distributional inequalities for noncommutative martingales

We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a method to build the noncommutative and classical probabilistic inequalities in an entirely operator theoretic way.     

Generalized kolmogorov inequalities for martingales

The classical ebyev inequality leads to an inequality for martingales which is often called the Kolmogorov inequality. It is shown here that many generalized ebyev inequalities for random variables lead in a similar way to martingale inequalities, and that the corresponding martingale inequality is sharp when the ebyev inequality is.On leave from Tel-Aviv University. Presently at the University of California, Berkeley.Research supported by National Science Foundation Grant MPS75-06173     

On Bernstein-type inequalities for martingales

Bernstein-type inequalities for local martingales are derived. The results extend a number of well-known exponential inequalities and yield an asymptotic inequality for a sequence of asymptotically continuous martingales.     

Weak type inequalities for vector-valued martingales

We obtain weak versions of the Burkholder–Davis–Gundy inequalities for vector-valued martingales by employing the operator-valued martingale transform technique. These inequalities are closely related to the geometric properties of the underlying Banach space.     

Some inequalities for martingales and stochastic convolutions

We give a stopped Doob inequality for a right continuous martingale in Hilbert space,, Using this we obtain inequalities for p-th moments with 0 < p < in terms of the Meyer process and the quadratic variation of the pure jump part. We also consider the convolution of a contraction type semigroup and a right continuous martingale and obtain inequalities similar to those of a martingale     

Rosenthal type inequalities for martingales in symmetric spaces

We establish inequalities similar to classical Rosenthal ones for sequences of martingale differences in general symmetric spaces. A central role is played by a predictable square variation of a martingale.     

Pointwise inequalities for ergodic averages and reversed martingales

It is known that the ergodic averages A_n? in the context of the shift action on satisfy pointwise inequalities of the form

     

On the maximal inequalities for martingales involving two functions

Let and be nonnegative convex functions, and let and be the right continuous derivatives of and respectively. In this paper, we prove the equivalence of the following three conditions: (i) (ii) and (iii) s_0,$">where and are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities.

     

Random martingales and localization of maximal inequalities

Let (X,d,μ) be a metric measure space. For ∅≠R⊆(0,∞) consider the Hardy-Littlewood maximal operator

     

Square integrable martingales orthogonal to every stochastic integral

Examples of square integrable martingales adapted to processes with independent increments and orthogonal to all stochastic integrals are constructed. If every square integrable martingale adapted to a process with stationary independent increments is a stochastic integral it is shown that the process must be a Wiener process.     

Weighted integral inequalities for the maximal geometric mean operator in martingales

Under appropriate conditions on Young's functions Φ₁ and Φ₂, we give necessary and sufficient conditions in order that weighted integral inequalities hold for the maximal geometric mean operator G in martingale Orlicz classes. When Φ₁=tp and Φ₂=tp, the inequalities revert to the ones of strong or weak (p,p)-type in martingale spaces.     

A non-commutative formula for the colored Jones function

The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation value of Wilson loops of a 3-dimensional gauge theory, the Chern–Simons theory. We present the colored Jones function as an evaluation of the inverse of a non-commutative fermionic partition function. This result is in the form familiar in quantum field theory, namely the inverse of a generalized determinant. Our formula also reveals a direct relation between the Alexander polynomial and the colored Jones function of a knot and immediately implies the extensively studied Melvin–Morton–Rozansky conjecture, first proved by Bar–Natan and the first author about 10 years ago. Our results complement recent work of Huynh and Le, who also give a non-commutative formulae for the colored Jones function of a knot, starting from a non-commutative formula for the R matrix of the quantum group ; see Huynh and Le (in math.GT/0503296).     

Prophet regions and sharp inequalities for pth absolute moments of martingales

Exact comparisons are made relating E|Y₀|^p, E|Y_n−1|^p, and E(max_j≤n−1 |Y_j|^p), valid for all martingales Y₀,…,Y_n−1, for each p ≥ 1. Specifically, for p > 1, the set of ordered triples {(x, y, z) : X = E|Y₀|^p, Y = E |Y_n−1|^p, and Z = E(max_j≤n−1 |Y_j|^p) for some martingale Y₀,…,Y_n−1} is precisely the set {(x, y, z) : 0≤x≤y≤z≤Ψ_n,p(x, y)}, where Ψ_n,p(x, y) = xψ_n,p(y/x) if x > 0, and = a_n−1,py if x = 0; here ψ_n,p is a specific recursively defined function. The result yields families of sharp inequalities, such as E(max_j≤n−1 |Y_j|^p) + ψ_n,p^*(a) E |Y₀|^p ≤ aE |Y_n−1|^p, valid for all martingales Y₀,…,Y_n−1, where ψ_n,p^* is the concave conjugate function of ψ_n,p. Both the finite sequence and infinite sequence cases are developed. Proofs utilize moment theory, induction, conjugate function theory, and functional equation analysis.