Monte-Carlo approximation of minimum entropy measures

Let mu (n) = 1/n Sigma (n)(i=1) deltax(i) denote the empirical measure asso ciated with a sequence (X-i)(i greater than or equal to1) of r.v. i.i.d. ac cording to a probability measure mu on a Polish space S. We give a necessar y and sufficient condition for the a.s. existence of N such that. for all n greater than or equal to N, there is a probability measure v(n) which mini mizes the relative entropy with respect to mu (n) under the constraint inte gral (S) f dv(n) = 0, where f : S --> R-d. Under this condition, we prove t hat a.s. v(n) converges weakly to the generalized solution v of the minimiz ation of the entropy with respect to mu constrained problem. (C) 2001 Acade mie des sciences/Editions scientifiques et medicales Elsevier SAS.