On the convergence of a bounded amart and a conjecture of Chatterji

Through the decomposition theorem of Lebesgue and Darst it is possible to define a generalized Radon-Nikodym derivative of a bounded additive set function with respect to a bounded countably additive set function. For a bounded amart the derivatives of the components are shown to converge almost everywhere. This result, together with a characterization of amarts, yields a theorem stated by Chatterji whose proof is incorrect.     

On the value of a stopped set function process

For certain types of stochastic processes {X_n | n ∈ $\text{N}$}, which are integrable and adapted to a nondecreasing sequence of σ-algebras $\text{F}$_n on a probability space (Ω, $\text{F}$, P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {$\text{F}$_n | n ∈ $\text{N}$}, when is$\text{sup}{}_{\mathrm{s}}\text{∫}{}_{\mathrm{\Omega }}\text{| X}{}_{\mathrm{\sigma }}\text{| dP}$finite, and when is there a stopping time $\text{σ}$ for which this supremum is attained? In the present paper we set the problem in a measure theoretic framework. This approach turns out to be fruitful since it reveals the root of the problem: It avoids the use of such notions as probability, null set, integral, and even σ-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research.     

Operator martingale decompositions and the Radon-Nikodým property in Banach spaces

We consider submartingales and uniform amarts of maps acting between a Banach lattice and a Banach lattice or a Banach space. In this measure-free setting of martingale theory, it is known that a Banach space Y has the Radon-Nikodým property if and only if every uniformly norm bounded martingale defined on the Chaney-Schaefer l-tensor product , where E is a suitable Banach lattice, is norm convergent. We present applications of this result. Firstly, an analogues characterization for Banach lattices Y with the Radon-Nikodým property is given in terms of a suitable set of submartingales (supermartingales) on . Secondly, we derive a Riesz decomposition for uniform amarts of maps acting between a Banach lattice and a Banach space. This result is used to characterize Banach spaces with the Radon-Nikodým property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent. In the case 1<p<∞, our results yield Lp(μ,Y)-space analogues of some of the well-known results on uniform amarts in L¹(μ,Y)-spaces.     

Amarts on Riesz Spaces

The concepts of conditional expectations, martingales and stopping times were extended to the Riesz space context by Kuo, Labuschagne and Watson （Discrete time stochastic processes on Riesz spaces, Indag. Math.,15（2004）, 435-451）. Here we extend the definition of an asymptotic martingale （amart） to the Riesz spaces context, and prove that Riesz space amarts can be decomposed into the sum of a martingale and an adapted sequence convergent to zero. Consequently an amart convergence theorem is deduced.     

B值一致渐近鞅强大数定律的一点注记

本利用了Ｂ值一致渐近鞅的Ｄｏｏｂ分解,对Ｂ值一致渐近鞅的收敛性作进一步的探讨,得到了Ｂ值一致渐近鞅的强大数定律的几个重要结果,从而将实值一致渐近鞅的强大数定律的一些结果推广到了Ｂ值一致渐近鞅的情形。     

Hyperamarts: Conditions for regularity of continuous parameter processes

The regularity of trajectories of continuous parameter process (X_t)_t∈R+ in terms of the convergence of sequence E(X_Tn) for monotone sequences (T_n) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (X_n)_1≤n≤∞ of integrable random variables, lim X_n exists and is equal to X_∞ and (X_T) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(X_Tn) = E(X_∞) for every increasing sequence (T_n) of extended simple stopping times converging to ∞. By applying these discrete parameter theorems, convergence theorems about continuous parameter processes are obtained. For example, it is shown that a progressive, optionally separable process (X_t)_t∈R+ with E{X_T} < ∞ for every bounded stopping time T is right continuous if lim E(X_Tn) = E(X_T) for every bounded stopping time T and every descending sequence (T_n) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained.