## The Annals of Probability

- Ann. Probab.
- Volume 23, Number 1 (1995), 420-445.

### Moderate Deviations of Dependent Random Variables Related to CLT

#### Abstract

This paper consists of three-parts. In the first-part, we find a common condition-the $C^2$ regularity--both for CLT and for moderate deviations. We show that this condition is verified in two important situations: the Lee-Yang theorem case and the FKG system case. In the second part, we apply the previous results to the additive functionals of a Markov process. By means of Feynman-Kac formula and Kasto's analytic perturbation theory, we show that the Lee-Yang theorem holds under the assumption that 1 is an isolated, simple and the only eigenvalue with modulus 1 of the operator $P_1$ acting on an appropriate Banach space $(b\mathscr{E}, C_b(E), L^2 \cdots)$. The last part is devoted to some applications to statistical mechanical systems, where the $C^2$-regularity becomes a property of the pressure functionals and the two situations presented above become exactly the Lee-Tang theorem case and the FKG system case. We shall discuss in detail the ferromagnetic model and give some general remarks on some other models.

#### Article information

**Source**

Ann. Probab., Volume 23, Number 1 (1995), 420-445.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988393

**Digital Object Identifier**

doi:10.1214/aop/1176988393

**Mathematical Reviews number (MathSciNet)**

MR1330777

**Zentralblatt MATH identifier**

0828.60017

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 60F05: Central limit and other weak theorems 60J25: Continuous-time Markov processes on general state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Moderate deviation CLT LIL Markov process analytic perturbation Gibbs measure Lee-Yang theory FKG and GHS inequalities

#### Citation

Liming, Wu. Moderate Deviations of Dependent Random Variables Related to CLT. Ann. Probab. 23 (1995), no. 1, 420--445. doi:10.1214/aop/1176988393. https://projecteuclid.org/euclid.aop/1176988393