Importance Sampling for multi-constraints rare event probability

Improving Importance Sampling estimators for rare event probabilities requires sharp approx- imations of the optimal density leading to a nearly zero-variance estimator. This paper presents a new way to handle the estimation of the probability of a rare event defined as a finite intersection of subset. We provide a sharp approximation of the density of long runs of a random walk condi- tioned by multiples constraints, each of them defined by an average of a function of its summands as their number tends to infinity.Comment: Conference pape

Some recent developments in quantization of fractal measures

We give an overview on the quantization problem for fractal measures, including some related results and methods which have been developed in the last decades. Based on the work of Graf and Luschgy, we propose a three-step procedure to estimate the quantization errors. We survey some recent progress, which makes use of this procedure, including the quantization for self-affine measures, Markov-type measures on graph-directed fractals, and product measures on multiscale Moran sets. Several open problems are mentioned.Comment: 13 page

Large Deviations Theory: Basic Principles and Applications to Communication Networks

The theory of large deviations refers to a collection of techniques for estimating properties of rare events such as their frequency and most likely manner of occurrence. Loosely speaking, LDT can be seen as a refinement of the classical limit theorems of probability theory and it is useful when simulation or numerical techniques become increasingly difficult as a parameter of interest tends to its limit. The first part of this tutorial deals with the behaviour of the empirical mean of IID RVs, the most natural framework to introduce the basic concepts and theorems of LDT and to highlight their heuristic interpretation. Then, the large deviation principle for the single server queue is presented and its implications on network dimensioning are discussed. Finally, the tutorial overviews the application of LDT to rare event simulation, for the choice of the optimal change of measure in Importance Samplin