The Annals of Probability

Moderate Deviations of Dependent Random Variables Related to CLT

Wu Liming

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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

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

First available in Project Euclid: 19 April 2007

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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]

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


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

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