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Principle of Conditioning revisited

Jakubowski A.

Demonstratio Mathematica

2012

Vol. 45, nr 2

325-336

Principle of Conditioning is a well known heuristic rule which allows constructing limit theorems for sums of dependent random variables from existing limit theorems for independent summands. In the paper we state a general limit theorem on converegnce to stable laws, which is valid for stationary sequences and provides a link between the Principle of Conditioning and ergodic theorems.

Random partition and stochastic integration in finite von Neumann algebras

Mohammed A. A. A.

Journal of Applied Analysis

2007

Vol. 13, nr 1

33-56

We introduce the notion of a random partition of the stochastic interval [τ 0, τ ∞] as an analogy to the classical case and characterize the predictable processes associated with such partitions. Also we identify the operator algebras connected with the stochastic integrals of predictable processes and examine their mutual relations.

Towards a general Doob-Meyer decomposition theorem

Jakubowski A.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

143--153

Both the Doob-Меуег and Graversen-Rao decomposition theorems can be proved following an approach based on predictable compensators of discretizations and weak-L1 technique, which was developed by K. M. Rao. It is shown that any decomposition obtained by Rao’s method gives predictability of compensators without additional assumptions (like submartingality in the original Doob-Meyer theorem or finite energy in the Graversen-Rao theorem).