Nonuniqueness in \(g\)-functions. (English) Zbl 0786.60043

Let \((X_ n)\) be a one-sided shift with a finite state space \(A\). The conditional expectation \(P(X_ 0=a_ 0 \mid X_{-1}=a_{-1},\;X_{- 2}=a_{-2},\dots)\) gives a function \(f(a_ 0 \mid a_{-1},a_{- 2},\dots)\) with \[ 0 \leq f(a_ 0 \mid a_{-1},a_{-2},\dots) \leq 1, \quad \sum_{a \in A} f(a \mid a_{-1},a_{-2},\dots)=1. \] Such functions are called \(g\)-functions; they were introduced by W. Doeblin and R. Fortet [Bull. Soc. Math. Fr. 65, 132-148 (1937; Zbl 0018.03303)]. To a \(g\)-function there might exist a corresponding stationary process, i.e. a (shift-)invariant measure on \(A^ N\). M. Keane [Invent. Math. 16, 309-324 (1972; Zbl 0241.28014); for more results see also H. Berbee, Probab. Theory Relat. Fields 76, 243-253 (1987; Zbl 0611.60059)] showed that this is the case for continuous \(g\)-measures and gave conditions for which the measure is strongly mixing and unique. B. Petit [C. R. Acad. Sci., Paris, Sér. A 280, 17-20 (1975; Zbl 0301.28012)] showed that all differentiable \(g\)-functions \(f\) with \[ \varepsilon \leq f \leq 1-\varepsilon \tag{*} \] for some \(0<\varepsilon<1/2\) have unique measures which are weakly Bernoulli. S. Kalikow [Isr. J. Math. 71, No. 1, 33-54 (1990; Zbl 0711.60041)] noticed that the continuity of a \(g\)-function \(f\) is equivalent to uniform convergence of the martingales \(P(X_ 0=a_ 0 \mid X_{- 1},X_{-2},\dots,X_{-n})\), \(n=1,2,\dots,\) \(a_ 0 \in A\), where \((X_ i)\) is the corresponding stationary process. The martingales converge uniformly if and only if \((X_ i)\) is a random Markov chain, i.e. there exist r.v. \(N_ i\) with values in \(\mathbb{N}\) such that \((X_ i,N_ i)\) is a stationary process, \(N_ 0\) is independent of \((a_ i,N_ i)_{i<0}\) and \[ P(X_ 0=a_ 0 \mid X_{-1}=a_{-1},\dots,X_{-n}=a_{-n},N_ 0=n)=P(X_ 0=a_ 0 \mid (X_ i)_{i<0}= (a_ i)_{i<0}, N_ 0=n) \tag{**} \] for all \(n\), \((a_ i)_{i<0}\).
From the proof of Theorem 7 of the Kalikow’s paper one can derive that if a continuous \(g\)-function \(f\) satisfies (*) and (**) with \(EN_ 0<\infty\), then the corresponding invariant measure is determined uniquely. The paper under review gives an example of a continuous \(g\)- function which satisfies (*) but has two measures.
Reviewer: D.Volný (Praha)


60G10 Stationary stochastic processes
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