## The Annals of Probability

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

Wu Liming

#### 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

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