Improving Importance Sampling estimators for rare event probabilities requires sharp approximations of conditional densities. This is achieved for events $E_{n}:=\left( u(X_{1})+...+u(X_{n}\right) )\in A_{n}$ 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 vector $\left( X_{1},...,X_{k_{n}}\right) $ with respect to $E_{n}$ on long runs, when $% k_{n}/n\rightarrow1$, is handled. The maximal value of $k_{n}$ compatible with a given accuracy is discussed; simulated results are presented, which enlight the gain of the present approach over classical IS schemes. Detailed algorithms are proposed.
