Improving Importance Sampling estimators for rare event probabilities
requires sharp approximations of conditional densities. This is achieved for
events E_{n}:=(f(X_{1})+...+f(X_{n}))\inA_{n} where the summands are i.i.d. and
E_{n} is a large or moderate deviation event. The approximation of the
conditional density of the real r.v's X_{i} 's, for 1\leqi\leqk_{n} with repect
to E_{n} on long runs, when k_{n}/n\to1, is handled. The maximal value of k
compatible with a given accuracy is discussed; algorithms and simulated results
are presented