In this paper we show that the local limit theorem is a consequence of the integral central limit theorem in the case of a Gibbs random field
t
,
t
Z
corresponding to a finite range potential.
We apply this theorem to show that the equivalence between Gibbs and canonical ensemble is a consequence of the integral central limit theorem and of very weak conditions on decrease of correlations.
Communicated by G. Gallavotti
Research supported by a C.N.R.-Ak. Nauk. U.R.S.S. fellowship