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Ergodicity of certain Markov chains and iterated function systems (Ergodicity)

Date: 1.1.1994-31.12.1998
Code: K6615
Department: Åbo Akademi University / Faculty of Mathematics and Natural Sciences (MNF), Dept. of Matehematics
Address: Fänriksgatan 3 B II vån, FIN-20500 Åbo
Phone +358-2-2154 224
Fax +358-2-2154 865
E-mail goran.hognas@abo.fi
Project leader: FD Göran Högnäs, professor (1.1.1994-31.12.1998)
Type of research: 0 (0=Within duty, 1=Ordered research, 2=Co-operation)
- basic research 100 %
Man months: Totally: 30 months
Contacts: Erasmus-nätverk (Twente); bilaterala (Dresden, Umeå); nationella (Helsingfors universitets Graduate School in analysis and logic och projektet Stokastiset prosessit ja niiden sovellukset).
Keywords: matematiska teorier, matemaattiset teoriat, entydigt invariant mått, ickelinjära autoregressiva processer, Eltons ergodteorem, svag konvergens, unique invariant measure, nonlinear autoregressive processes, Elton's ergodic theorem, weak convergence,

Consider the following general discrete-time Markov chain model on some topological space E (in most applications a subset of /Re^^): (1) X_=f(X_,/varepsilon_), n=0,1,2,...where f is a function from E x F to E and the sequence /varepsilon_, n=1,2,... taking values in some topological parameter space F is stationary and indipendent of the past X's.The aim of the research project is to develop verifiable suffucient conditions - preferably directly in terms of the one-step transition probalities - for positive recurrence and ergodicity of the process X.The key idea is to use drift conditions, or stochastic Lyapunov functions, to establish ergodicity for the stochastic variants of some widely used deterministic models.Applications are drawn, e.g., from discretization error analysis in engineering and simple stochastic population models in theoretically biology as well as computer graphics (fractal generation and iterated function systems).There is a definitive need to work out a rigorous mathematical basis in term of ergodic theorems for the simulation procedures which are extensively used to study, experimentally, the asymptotics of non-linear processes of the general form (1).

Publication(s)

26.3.1996 / 27.3.1996