## 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)

### MSC:

 60G10 Stationary stochastic processes
Full Text:

### References:

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