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


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
Full Text: DOI


[1] Arratia, R.: The motion of a tagged particle in the simple symmetric exclusion system in Z. Ann. Probab.11 362-373 (1983) · Zbl 0515.60097 · doi:10.1214/aop/1176993602
[2] Helland, I.: Central limit theorems for martingales with discrete or continuous time. Scand. J. Stat.,9, 79-94 (1982) · Zbl 0486.60023
[3] Kipnis, C., Lebowitz, J. L., Presutti, E., Spohn, H.: J. Stat. Phys.30, 107-121 (1983) · doi:10.1007/BF01010870
[4] Lebowitz, J. L., Spohn, H.: J. Stat. Phys.28, 539-556 (1982) · Zbl 0512.60075 · doi:10.1007/BF01008323
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.