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On limit theorems in JW-algebras

Karimov A., Mukhamedov F.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 1

153--165

In the present paper, we study bundle convergence in JW- algebra and prove certain ergodic theorems with respect to such convergence. Moreover, conditional expectations of reversible JW-algebras are considered. Using such expectations, the convergence of supermartingales is established.

On the almost sure convergence of all conditionings of positive random variables

Paszkiewicz A.

Demonstratio Mathematica

2001

Vol. 34, nr 2

345-352

Ergodic theorems for Markov chains represented by iterated function systems

Stenflo O.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 1

27-43

We consider Markov chains represented in the form Xn+1 = f (Xn, In), where {In} is a sequence of independent, identically distributed (i.i.d.) random variables, and where f is a measurable function. Any Markov chain {Xn} on a Polish state space may be represented in this form i.e. can be considered as arising from an iterated function system (IFS). A distributional ergodic theorem, including rates of convergence in the Kantorovich distance is proved for Markov chains under the condition that an IFS representation is "stochastically contractive" and "stochastically bounded". We apply this result to prove our main theorem giving upper bounds for distances between invariant probability measures for iterated function systems. We also give some examples indicating how ergodic theorems for Markov chains may be proved by finding contractive IFS representations. These ideas are applied to some Markov chains arising from iterated function systems with place dependent probabilities.

A non-ergodic phenomenon for some random dynamical system

Komisarski A.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 2

407--412

In [2] Jajte formulated the following question: Let h0 (x) and h1 (x) be homeomorphisms of the interval [0, 1] onto itself. Is it true that for any x € [0, 1] and almost any t ϵ (0, 1) there exists a limit of a sequence [formula] for n → ∞, where t = (0, t1 t2…)2 is a binary representation of t, i.e. t = Σi ≥ 1 ti 2-i and ti ϵ {0, 1}? The answer is negative. We describe the set of condensation points of the sequence in some special cases.