The Adaptive Multilevel Splitting algorithm is a very powerful and versatile iterative method to estimate the probability of rare events, based on an interacting particle systems. In an other article, in a so-called idealized setting, the authors prove that some associated estimators are unbiased, for each value of the size n of the systems of replicas and of resampling number k. Here we go beyond and prove these estimator's asymptotic normality when h goes to infinity, for any fixed value of k. The main ingredient is the asymptotic analysis of a functional equation on an appropriate characteristic function. Some numerical simulations illustrate the convergence to rely on Gaussian confidence intervals.