Moderate Deviations of Dependent Random Variables Related to CLT

Liming, Wu

Ann. Probab., Tome 23 (1995) no. 3, p. 420-445 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1995-01-14
Classification:  Moderate deviation,  CLT,  LIL,  Markov process,  analytic perturbation,  Gibbs measure,  Lee-Yang theory,  FKG and GHS inequalities,  60F10,  60F05,  60J25,  60K35