Approximation of measures by Markov processes and homogeneous affine iterated function systems
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It is shown that under certain conditions, attractive invariant measures for iterated function systems (indeed for Markov processes on locally compact spaces) depend continuously on parameters of the system.
We discuss a special class of iterated function systems, the homogeneous affine ones, for which an inverse problem is easily solved in principle by Fourier transform methods. We show that a probability measureμ onR n can be approximated by invariant measures for finite iterated function systems of this class if\(\hat \mu (t)/\hat \mu (a^T t)\) is positive definite for some nonzero contractive linear mapa:R n →R n . Moments and cumulants are also discussed.
AMS classification60J05 60J15 60F05 42A38 42A82 58F11 41A99
Key words and phrasesInvariant measures Markov processes Iterated function systems Approximation of measures Fourier transform Positive definite
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