Martingale convergence via the square function

By exploiting the natural setting of a convergence theorem of Burkholder, a direct and elementary proof of the theorem is given. This proof is also new for the martingale convergence theorem and the martingale transform convergence theorem which are corollaries to the above-mentioned convergence theorem.

CHAPTER 7 - Conditional Expectation and Martingale Theory

International audience

The martingale convergence theorem is first proved for uniformly integrable martingales by a standard application of Doob’s maximal inequality. A simple truncation argument is then given which reduces the proof of the L 1 {L^1} -bounded martingale theorem to the uniformly integrable case. A similar method is used to prove Burkholder’s martingale transform convergence theorem.

In Chapter 17 we studied convergence theorems, but they were all of the type that one form of convergence, plus perhaps an extra condition, implies another type of convergence. What is unusual about martingale convergence theorems is that no type of convergence is assumed — only a certain structure — yet convergence is concluded. This makes martingale convergence theorems special in analysis; the only similar situation arises in ergodic theory.

On abstract conditional expectations

By using Doob's martingale convergence theorem, this paper presents a class of strong limit theorems for arbitrary stochastic sequence. Chow's two strong limit theorems for martingale-difference sequence and Loeve's and Petrov's strong limit theorems for independent random variables are the particular cases of the main results.

A novel, tabu-based real-coded small-world optimization algorithm (TR-SWA) is proposed. Tabu search is adopted to avoid duplicate searches of the real-coded small-world optimization algorithm (R-SWA). A crossover operator is introduced to construct search operators. The convergence behaviour of this TR-SWA scheme is shown by establishing the Markov model, and it is proved that TR-SWA meets the convergence theorem of a general random search algorithm proposed by Solis and Wets. Simultaneously, martingale convergence theorems are used to prove the nearly universal strong convergence of TR-SWA. Finally, five benchmark functions are introduced to evaluate the performance of TR-SWA: comparisons are made between TR-SWA, particle swarm optimization, binary-coded small-world optimization algorithm and R-SWA. Numerical experiments demonstrate that the addition of the tabu search improves the performance of R-SWA for most of the investigated optimization problems, and the global convergence of TR-SWA is guaranteed if the feasible set is bounded.

Let F be a mapping from real n -dimensional Euclidean space into itself. Most practical algorithms for finding a zero of F are of the form \[ x k + 1 = x k − B k − 1 F x k , {x_{k + 1}} = {x_k} - B_k^{ - 1}F{x_k}, \] where { B k } \{ {B_k}\} is a sequence of nonsingular matrices. The main result of this paper is a characterization theorem for the superlinear convergence to a zero of F of sequences of the above form. This result is then used to give a unified treatment of the results on the superlinear convergence of the Davidon-Fletcher-Powell method obtained by Powell for the case in which exact line searches are used, and by Broyden, Dennis, and Moré for the case without line searches. As a by-product, several results on the asymptotic behavior of the sequence { B k } \{ {B_k}\} are obtained. An interesting aspect of these results is that superlinear convergence is obtained without any consistency conditions; i.e., without requiring that the sequence { B k } \{ {B_k}\} converge to the Jacobian matrix of F at the zero. In fact, a modification of an example due to Powell shows that most of the known quasi-Newton methods are not, in general, consistent. Finally, it is pointed out that the above-mentioned characterization theorem applies to other single and double rank quasi-Newton methods, and that the results of this paper can be used to obtain their superlinear convergence.

The representation Skorohod theorem of weak convergence of random variables on a metric space goes back to Skorohod (1956) in the case where the metric space is the class of real-valued functions defined on [0,1] which are right-continuous and have left-hand limits when endowed with the Skorohod metric. Among the extensions of that to metric spaces, the version by Wichura (1970) seems to be the most fundamental. But the proof of Wichura seems to be destined to a very restricted public. We propose a more detailed proof to make it more accessible at the graduate level. However we do far more by simplifying it since important steps in the original proof are dropped, which leads to a direct proof that we hope to be more understandable to a larger spectrum of readers. The current version is more appropriate for different kinds of generalizations.

In this paper, we shall first give a general method for convergence theorem of fuzzy set-valued random variables. By using this "sandwich" method, we give the proofs of convergence theorems for fuzzy set-valued martingales, sub- and supermartingales in the sense of the extended Hausdorff metric H∞. Also we shall state a convergence result about uniform amarts.

On the convergence of fuzzy martingales

Martingales and Ergodic Theory

In this paper, various convergence theorems and criteria of closedness of multivalued martingales, submartingales, and supermartingales are proved.

Global optimization in problems with uncertainties: the gamma algorithm