Martingale Convergence Research Articles

On abstract conditional expectations

(i) If C is a sub-a-field of 2, f,(o) -“f(o) p-a.e. and there is a nonnegative real valued function 40 E L’(Q) s.t. I/f,(o)\/ 1 then lim,,, Ezf,(w) = E”(o). This is just the Lebesgue dominated convergence theorem for abstract conditional expectations (see [S]). (ii) If {zm, z},, , are sub-a-fields of 2, V,“= , C, = Z or fi‘I,“=, 2, = C andfEL$(Q) (1 <PC co) then EZn -+L%(“) E’J: Furthermore if p = 1 then EZnf(o) --+’ Ezf(w) p-a.e. These are known as martingale convergence theorems (see [S] and [14, Theorem 51). Together with the general theory of martingales have an increasingly important impact in Banach space theory (see [5, Chap. VI).

On Strong Consistency of a Stochastic Approximation Algorithm for the Identification of Time-Varying Non-Minimum Phase Systems

Although most real systems are time-varying, few authors have analysed the behaviour of such systems. In this paper strong consistency results are developed for identification algorithms for a class of time varying-systems under minimal assumptions. The input is made persistently by the addition of an auxiliary signal, but it is not assumed that the system is stable or minimum phase. The analysis draws on convergence results for time-varying systems previously developed by the author and the extension of some well-known martingale convergence theorems.

On two classes of interacting stochastic processes arising in cancer modeling

The processes X and Y are said to interact if the laws governing the changes of either of them at time t depend on the values of the other process at times up to t. For bivariate interacting Markov processes, their limiting behavior is analysed by means of an approximation suggested by Fuhrmann, consisting of discretizing time, and assuming that in each time interval the processes develop independently, according to the laws obtained by fixing the value of the other process at its level attained at the beginning of the interval. In this way the conditions for almost sure extinction, escape to ∞ with positive probability, etc., are obtained (by using the martingale convergence theorem) for state-dependent branching processes studied by Roi, and for bivariate processes with one component piecewise determined.

On conditional optimality of a class of learning automata in random environments

In this paper, we develop a new class of learning automata to operate in a random environment with a finite number of responses, called a Q-model, and investigate its asymptotic behavior from the viewpoint of conditional optimality. It is shown that this class contains as a particular case the β-model automaton introduced by Luce, which is available for an environment with binary responses, called a P-model. Making use of the martingale convergence theorem for the asymptotic analysis of the automata, we show that the automata are conditionally optimal, i.e., the automata converge optimally under the restrictions on the probability distributions that characterize the environment. The usefulness of the result is illustrated by its application to two simple automata: the β-model automaton and another one which is also applied to a P-model environment. Then, extending the idea underlying the second automaton, we develop a design method for an effective and structurally simple automaton for any Q-model environment. The conditional optimality of the resulting automaton is also investigated.

Convergence of Martingales on a Riemannlan Manifold

When the scalar quadratic variation of a martingale on a Riemannian manifold is finite almost surely, then the martingale converges almost surely in the one-point compactification of the manifold. A partial converse due to Zheng Wei-an is also proved. No curvature conditions on the manifold are required.

Strong Convergence of Martingales in von Neumann Algebras

In this paper we prove strong and ${L^1}$-norm convergence of martingales with respect to a faithful normal semifinite weight on a von Neumann algebra.

ODE Method Versus Martingale Convergence Theory

During the last few years two methods, namely The Ordinary Differential Equation method, due to Ljung and Martingale convergence Theory which was used by several authors have been proposed for the convergence analysis of the processes governed by stochastic difference equations. The idea of the ODE method is to associate the set of stochastic difference equations governing the process dynamics with a set of ordinary differential equations whose stability properties contain the necessary information about the asymptotic convergence behaviour of the algorithm. An alternative analysis can be done by means of the Martingale convergence theory, what is a so called “stochastic version of the Ljapunov stability theorem”. Both methods are applicable to the stochastic convergence analysis of recursive parameter estimation methods and adaptive systems. The purpose of this work is to compare both methods with respect to the assumptions needed and to the conclusions obtained.

Weak convergence of martingales with random indices to infinitely divisible laws

Weak convergence of martingales with random indices to infinitely divisible laws

On one-parameter proofs of almost sure convergence of multiparameter processes

The martingale convergence theorem of R. Cairoli [-4] was proved assuming that the filtration satisfies a conditional independence assumption usualy called (F4). It belongs to the folklore of the subject (see also [-17]) that (F4) can be restated as a condition on commutation of conditional expectation operators. This formulation will allow us to derive Cairoli's theorem from a simple general argument about operators on Orlicz spaces. The advantage is that one obtains a unified proof of multi-parameter versions of several other results: the theorems of Rota, Dunford-Schwartz, Akcoglu, and Stein. H. FNlmer was able to apply Proposition 2.1 below to random fields.

Strong convergence of martingales in von Neumann algebras

In this paper we prove strong and L 1 {L^1} -norm convergence of martingales with respect to a faithful normal semifinite weight on a von Neumann algebra.

Global convergence of output error recursions in colored noise

This paper presents a variation on a known extended least squares algorithm of the output or parallel type. Under reasonable conditions, the algorithms achieve global convergence of the one-step-ahead prediction error to the additive independent (possible colored) measurement noise. The convergence of the algorithms proposed is not critically sensitive to the color in the noise as are related extended least squares schemes which require a simultaneous noise model identification, nor is the convergence critically sensitive to the input signals as are realizations of the method of instrumental variables. The algorithms are also simpler to implement than for the competing schemes. In the paper, there is also studied an add on scheme which consists of additional processing of the prediction errors to achieve simultaneous noise model identification, and improved prediction. Such a scheme is attractive from the computational cost point of view. Global convergence results are developed for the algorithms based on martingale convergence theorems as in earlier theories for extended least squares schemes. The key contribution of the paper as far as the theory is concerned is to show how to cope with the colored noise in the martingale framework.

Martingale convergence analysis of adaptive schemes--A feedback approach

An equivalent stochastic feedback system can be associated to many recursive identification and adaptive control schemes. This equivalent feedback system has a certain structure which can be used for selecting appropriate near supermartingales allowing us to establish a.s. convergence. A basic convergence result is presented which can be used in the analysis of various adaptive schemes.

Martingale Convergence Analysis of Output Error Identification Algorithms

Martingale convergence analysis of two output error recursive iden-tification schemes is presented. The analysis is carried on by associating to each scheme an equivalent stochastic feedback system. These equivalent stochastic feedback systems belong to a general class of feedback systems for which explicit convergence conditions can be established using a martingale approach. The results obtained with this approach confirms previous results obtained using O.D.E. but they are stronger since it is shown that monitoring is not necessary.

A nonstandard proof of the Martingale convergence theorem

A nonstandard proof of the Martingale convergence theorem

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.

An invariance principle for solutions to stochastic difference equations

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.

An invariance principle for solutions to stochastic difference equations

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.

The Order Convergence of Martingales Indexed by Directed sets

We obtain a condition on the underlying family of $\sigma$-algebras which is properly weaker than the Vitali property but which is also a sufficient condition for the order convergence of martingales of semibounded variation. We also obtain a sufficient condition for the order convergence of martingales of semibounded variation in terms of the finiteness of the extreme order limits of martingales of bounded variation.

Order Convergence of Martingales in Terms of Countably Additive and Purely Finitely Additive Martingales

Let $(E, \mathscr{B}, \mu)$ be a measure space, let $\theta$ be a directed set with a countable cofinal subset, and let $(\mathscr{B}_\tau)_{\tau \in \theta}$ be an increasing family of sub-$\sigma$-algebras of $\mathscr{B}$. A martingale $(f_\tau)_{\tau \in \theta}$ is said to be of semibounded variation whenever the set $\{\int_B f_\tau d\mu \mid \tau \in \theta, B \in \mathscr{B}_\tau\}$ is bounded either from above or below. Denote conditional expectation by $\mathscr{E}$. We show that if every martingale of the form $\mathscr{E}(f \mid \mathscr{B}_\tau)_{\tau\in\theta}$ for some $\mathscr{B}$-measurable function $f$ with $\int|f| d\mu < \infty$ is order convergent, then every martingale of semibounded variation is order convergent. When the family $(\mathscr{B}_\tau)_{\tau\in\theta}$ satisfies a certain refinement condition, we obtain a sufficient condition for order convergence of martingales of semibounded variation in terms of order convergence of martingales which converge stochastically to 0.

Best ?- and N ?-approximants in Orlicz spaces of vector valued functions

Let (Ω, A, P) be a probability space and E be a Banach space. We study the approximation of an E-valued random variable X, which is an element of the Orlicz space LΦ(Ω, A, P; E), by a function Y∃LΦ, which is measurable with respect to a sub-σ-field of A and takes values in a closed convex subset of E. Two types of approximation are considered: ∫Φ(∥X − Y∥) dP=inf, and NΦ(X−Y)=inf with the Orlicz space norm NΦ. We give conditions for the existence of best approximants. If E is reflexive, we obtain martingale type convergence theorems for best approximants and discuss the continuity of the operator X → best approximant of X.