### Abstract

We study the asymptotic behavior of partial sums S_{n}for certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number λ, and a positive integer k so that (S_{n}-nm)/n^{1-1/2k} converges weakly to a random variable with density proportional to exp(-λ|s|^{2k}/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.

Original language | English (US) |
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Pages (from-to) | 117-139 |

Number of pages | 23 |

Journal | Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete |

Volume | 44 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1978 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Analysis
- Mathematics(all)

### Cite this

**Limit theorems for sums of dependent random variables occurring in statistical mechanics.** / Ellis, Richard S.; Newman, Charles M.

Research output: Contribution to journal › Article

*Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete*, vol. 44, no. 2, pp. 117-139. https://doi.org/10.1007/BF00533049

}

TY - JOUR

T1 - Limit theorems for sums of dependent random variables occurring in statistical mechanics

AU - Ellis, Richard S.

AU - Newman, Charles M.

PY - 1978/6

Y1 - 1978/6

N2 - We study the asymptotic behavior of partial sums Snfor certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number λ, and a positive integer k so that (Sn-nm)/n1-1/2k converges weakly to a random variable with density proportional to exp(-λ|s|2k/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.

AB - We study the asymptotic behavior of partial sums Snfor certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number λ, and a positive integer k so that (Sn-nm)/n1-1/2k converges weakly to a random variable with density proportional to exp(-λ|s|2k/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.

UR - http://www.scopus.com/inward/record.url?scp=0008999076&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0008999076&partnerID=8YFLogxK

U2 - 10.1007/BF00533049

DO - 10.1007/BF00533049

M3 - Article

VL - 44

SP - 117

EP - 139

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 2

ER -