## Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions.(English)Zbl 0588.60058

Let $$\{y_ j\}_{-\infty <j<\infty}$$ be an ergodic discrete parameter stationary process given by a reversible Markov chain on a state space (X,$${\mathcal X})$$ with transition probability q and invariant probability distribution $$\pi$$. Let V be a real valued function defined on X and assume that $\int V(x)\pi (dx)=0,\quad \int V^ 2(x)\pi (dx)<\infty,\quad and\quad \lim_{n\to \infty}n^{-1}E[V(y_ 1)+...+V(y_ n)]^ 2=\sigma^ 2<\infty.$ (This last condition is equivalent to $$V\in Range(I-\bar q)^{1/2}$$, where $$\bar q$$ denotes the transition operator associated with q and defined on real valued bounded $${\mathcal X}$$-measurable functions on X). Denote by $$F_ n$$ the $$\sigma$$- field generated by the $$y_ j$$ for $$j\leq n$$. It is proved that $$X_ n=\sum^{n}_{j=1}V(y_ j)$$ can be written as $$M_ n+\epsilon_ n$$, where $$M_ n$$ is a martingale relative to $$F_ n$$, $$n\geq 1$$, and $$\lim_{n\to \infty}\sup_{1\leq j\leq n}| \xi_ j| =0,$$ $$\lim_{n\to \infty}n^{-1}E\xi^ 2_ n=0.$$
It follows at once that $$X_ n$$ obeys the functional central limit theorem. An analogous result for continuous parameter processes is also stated. The results obtained are used in the study of the asymptotic behaviour of a tagged particle in an infinite particle system performing simple excluded random walk.
Reviewer: M.Iosifescu

### MSC:

 60J05 Discrete-time Markov processes on general state spaces 60J25 Continuous-time Markov processes on general state spaces 60F17 Functional limit theorems; invariance principles 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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### References:

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