Language:
EN
| Published:
30-09-2013
|
Abstract
| pp. 39-58
The paper describes briefly a history of filtering problems of Markov processes and then concentrates on ergodic properties of filtering process. A mistake in a famous Kunita paper on ergodicity of filtering processes is shown. Then the paper reviews various attempts trying to correct this mistake.
We use the approach from Czudek and Szarek (see [1]) to prove the central limit theorem for a stationary Markov chain generated by an iterative function system for a family of increasing, injective functions on [0, 1] with "contractive" properties. We introduce a new approach to prove existence of an unique invariant measure using e-property (see [2]).
Language:
EN
| Published:
20-05-2025
|
Abstract
| pp. 177-189
Let P be a Markov operator on a general state space (S, Σ) with an invariant probability measure m, assumed to be ergodic. We study conditions which yield that for every centered non-zero f ∈ L2(m) a non-degenerate annealed CLT and an L2-normalized CLT hold.
Language:
EN
| Published:
15-07-2024
|
Abstract
| pp. 134-154
The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota’s results on the author’s research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differentia equations.
Language:
EN
| Published:
23-09-2016
|
Abstract
| pp. 129-142
In this paper we want to show the existence of a form of asymptotic stability of random dynamical systems in the sense of L. Arnold using arguments analogous to those presented by T. Szarek in [6], that is showing it using conditions generalizing the notion of tightness of measures. In order to do that we use tightness theory for random measures as developed by H. Crauel in [2].
Language:
EN
| Published:
30-08-2021
|
Abstract
| pp. 236-249
In this paper our considerations are focused on some Markov chain associated with certain piecewise-deterministic Markov process with a statedependent jump intensity for which the exponential ergodicity was obtained in [4]. Using the results from [3] we show that the law of iterated logarithm holds for such a model.
2021-08-30
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