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A new class of Markov processes for image encoding. (English) Zbl 0643.60050

Let \(w_ i\), \(i=1,...,N\), be Lipschitz functions from X into X, (X,d) a complete metric space, with \(d(w_ ix,w_ iy)\leq s_ id(x,y)\) for all x,y\(\in X\). For probabilities \(p_ i>0\), \(i=1,...,N\), \(\sum p_ i=1\), a discrete-time Markov process arises in a natural way from iteratively applying the maps \(w_ i:\) \(P(x,B)=\sum p_ i\delta_{w_ ix}(B).\)
Theorem 1. Suppose there exists \(r<1\) such that for all x,y\(\in X\) \[ (*)\quad \prod^{N}_{i=1}d(w_ ix,w_ iy)^{p_ i}=r d(x,y). \] Then there exists a unique, attractive and invariant initial distribution for this Markov process.
This extends results for compact X and strongly contractive \(w_ i's\) [e.g. the first author and S. Demko, Proc. R. Soc. Lond., Ser. A 399, 243-275 (1985; Zbl 0588.28002)]. (*) is merely an “average contractivity between points” condition. For the \(w_ i's\) being affine maps on \(R^ d\), a weaker condition is needed (Corollary 2). Also conditions for the existence of various moments of the measure are given.
Proposition 2. Let \(X=R^ 1\), \(w_ 1x=x/2\), \(w_ 2x=x+1\), \(0<p_ i<1\). Then the invariant measure is concentrated on [0,\(\infty)\), singular with respect to Lebesgue measure, yet has a strictly increasing distribution function. (It seems that this is the simplest known example of such a measure.)
Reviewer: U.Zähle

MSC:

60J05 Discrete-time Markov processes on general state spaces
60G30 Continuity and singularity of induced measures
60D05 Geometric probability and stochastic geometry

Citations:

Zbl 0588.28002
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