### Abstract

Some invariance principles for additive functionals of simple exclusion with finite-range translation-invariant jump rates p(i, j) = p(j - i) in dimensions d ≥ 1 are established. A previous investigation concentrated on the case of p symmetric. The principal tools to take care of nonreversibility, when p is asymmetric, are invariance principles for associated random variables and a "local balance" estimate on the asymmetric generator of the process. As a by-product, we provide upper and lower bounds on some transition probabilities for mean-zero asymmetric second-class particles, which are not Markovian, that show they behave like their symmetric Markovian counterparts. Also some estimates with respect to second-class particles with drift are discussed. In addition, a dichotomy between the occupation time process limits in d = 1 and d ≥ 2 for symmetric exclusion is shown. In the former, the limit is fractional Brownian motion with parameter 3/4, and in the latter, the usual Brownian motion.

Original language | English (US) |
---|---|

Pages (from-to) | 277-302 |

Number of pages | 26 |

Journal | Annals of Probability |

Volume | 28 |

Issue number | 1 |

State | Published - Jan 2000 |

Externally published | Yes |

### Fingerprint

### Keywords

- Associated
- Central limit theorem
- FKG
- Invariance principle
- Second-class particles
- Simple exclusion process

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

**Central limit theorems for additive functionals of the simple exclusion process.** / Sethuraman, Sunder.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 28, no. 1, pp. 277-302.

}

TY - JOUR

T1 - Central limit theorems for additive functionals of the simple exclusion process

AU - Sethuraman, Sunder

PY - 2000/1

Y1 - 2000/1

N2 - Some invariance principles for additive functionals of simple exclusion with finite-range translation-invariant jump rates p(i, j) = p(j - i) in dimensions d ≥ 1 are established. A previous investigation concentrated on the case of p symmetric. The principal tools to take care of nonreversibility, when p is asymmetric, are invariance principles for associated random variables and a "local balance" estimate on the asymmetric generator of the process. As a by-product, we provide upper and lower bounds on some transition probabilities for mean-zero asymmetric second-class particles, which are not Markovian, that show they behave like their symmetric Markovian counterparts. Also some estimates with respect to second-class particles with drift are discussed. In addition, a dichotomy between the occupation time process limits in d = 1 and d ≥ 2 for symmetric exclusion is shown. In the former, the limit is fractional Brownian motion with parameter 3/4, and in the latter, the usual Brownian motion.

AB - Some invariance principles for additive functionals of simple exclusion with finite-range translation-invariant jump rates p(i, j) = p(j - i) in dimensions d ≥ 1 are established. A previous investigation concentrated on the case of p symmetric. The principal tools to take care of nonreversibility, when p is asymmetric, are invariance principles for associated random variables and a "local balance" estimate on the asymmetric generator of the process. As a by-product, we provide upper and lower bounds on some transition probabilities for mean-zero asymmetric second-class particles, which are not Markovian, that show they behave like their symmetric Markovian counterparts. Also some estimates with respect to second-class particles with drift are discussed. In addition, a dichotomy between the occupation time process limits in d = 1 and d ≥ 2 for symmetric exclusion is shown. In the former, the limit is fractional Brownian motion with parameter 3/4, and in the latter, the usual Brownian motion.

KW - Associated

KW - Central limit theorem

KW - FKG

KW - Invariance principle

KW - Second-class particles

KW - Simple exclusion process

UR - http://www.scopus.com/inward/record.url?scp=0034336981&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034336981&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034336981

VL - 28

SP - 277

EP - 302

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 1

ER -