Introduction 1 1. Definition and estimation of Sobol indices 2 1.1. Exact model 2 1.2. Estimation of S 3 2. Asymptotic properties: exact model 4 2.1. Consistency and asymptotic normality 4 2.2. Asymptotic efficiency 6 3. Asymptotic properties: metamodel 8 3.1. Metamodel-based estimation 8 3.2. Consistency and asymptotic normality 8 3.3. Asymptotic efficiency 11 4. Numerical illustrations 12 4.1...

We consider conditioned Galton–Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. Another example is the number of protected nodes. The offs...

We consider conditioned Galton–Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. The offspring distribution defining the random tree is ass...

The look-ahead estimator is used to compute densities associated with Markov processes via simulation. We study a framework that extends the look-ahead estimator to a much broader range of applications. We provide a general asymptotic theory for the estimator, where both L1 consistency and L2 asymptotic normality are established. The L2 asymptotic normality implies √ n convergence rates for L2 ...

The Gaussian function G(x) = 1 p 21⁄4 e¡x 2=2; which has been a classical choice for multiscale representation, is the solution of the scaling equation G(x) = Z R ®G(®x¡ y)dg(y); x 2 R; with scale ® > 1 and absolutely continuous measure dg(y) = 1 p 21⁄4(®2 ¡ 1) e¡y 2=2(®2¡1)dy: It is known that the sequence of normalized B-splines (Bn); where Bn is the solution of the scaling equation Á(x) = n ...

Various types of graph statistics for Bernoulli graphs are represented as numerators of incomplete U-statistics. Asymptotic normality of these statistics is proved for Bernoulli graphs in which the edge probability is constant. In addition it is shown that subgraph counts asymptotically are linear functions of the number of edges in the graph. AMS Subject Classification: Primary 05C99; Secondar...