Analysis of adaptive multilevel splitting algorithms in an idealized case (Q2786485)

The paper investigates the adaptive multilevel splitting algorithms in the context of an idealized case. One aim of the paper is to discuss the necessity of the assumption on the continuity of the cumulative distribution function of a given a real-valued positive random variable \(X\). The authors prove that such an assumption is only required locally near \(0\). Denote for \(a>0\), \(p=p(X>a)\) and for integers \(n,k\) where \(n\) designs the total number of particles and \(k\) the number of resampled particles at each iteration. Consider also an estimator NEWLINE\[NEWLINE \hat{p}^{n,k}=C^{n,k}(1-\frac{k}{n})^{J^{n,k}} NEWLINE\]NEWLINE of the probability \(p\), and where \(C^{n,k}\) and \(J^{n,k}\) are appropriate constants depending on \(n\) and \(k\). The authors prove that \(\hat{p}^{n,k}\) is an unbiased estimator of \(p\) and obtain an explicit asymptotic expression for the variance of such an estimator. The crucial idea is based on adapting the family of conditional probabilities \(P(X > a\mid X > x)\) and to consider corresponding estimators relatively to the adaptive multilevel splitting algorithm. The paper is achieved by the analysis of the computational cost by deriving functional equations for the variance and the mean number of iterations in the algorithm.