### Abstract

We give easily verifiable conditions under which a functional central limit theorem holds for additive functionals of symmetric simple exclusion and symmetric zero-range processes. Also a reversible exclusion model with speed change is considered. Let η(t) be the configuration of the process at time t and let f(η) be a function on the state space. The question is: For which functions f does λ ^{-1/2}∫^{λt}_{0} f(η(s)) ds converge to a Brownian motion? A general but often intractable answer is given by Kipnis and Varadhan. In this article we determine what conditions beyond a mean-zero condition on f(η) are required for the diffusive limit above. Specifically, we characterize the H^{-1} space in an applicable way. Our method of proof relies primarily on a sharp estimate on the "spectral gap" of the process and weak regularity properties for the invariant measures.

Original language | English (US) |
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Pages (from-to) | 1842-1870 |

Number of pages | 29 |

Journal | Annals of Probability |

Volume | 24 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1996 |

Externally published | Yes |

### Keywords

- Central limit theorem
- Invariance principle
- Simple exclusion process
- Zero-range process

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Probability*,

*24*(4), 1842-1870. https://doi.org/10.1214/aop/1041903208