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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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