The estimation of rare event probability is a crucial issue in areas such as reliability, telecommunications, aircraft management. In complex systems, analytical study is out of question and one has to use Monte Carlo methods. When rare is really rare, which means a probability less than 10 −9 , naive Monte Carlo becomes unreasonable. A widespread technique consists in multilevel splitting, but this method requires enough knowledge about the system to decide where to put the levels at hand. This, unfortunately, is not always possible. In this article, we propose an adaptive algorithm to cope with this problem: The estimation is asymptotically consistent, costs just a little bit more than classical multilevel splitting, and has the same efficiency in terms of asymptotic variance. In the one-dimensional case, we rigorously prove the a.s. convergence and the asymptotic normality of our estimator, with the same variance as with other algorithms that use fixed crossing levels. In our proofs we mainly use tools from the theory of empirical processes, which seems to be quite new in the field of rare events.

International audienceThis paper discusses a novel strategy for simulating rare events and an associated Monte Carlo estimation of tail probabilities. Our method uses a system of interacting particles and exploits a Feynman-Kac representation of that system to analyze their fluctuations. Our precise analysis of the variance of a standard multilevel splitting algorithm reveals an opportunity for improvement. This leads to a novel method that relies on adaptive levels and produces, in the limit of an idealized version of the algorithm, estimates with optimal variance. The motivation for this theoretical work comes from problems occurring in watermarking and fin- gerprinting of digital contents, which represents a new field of applications of rare event simulation techniques. Some numerical results show performance close to the idealized version of our technique for these practical applications

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A method to generate reactive trajectories, namely equilibrium trajectories leaving a metastable state and ending in another one is proposed. The algorithm is based on simulating in parallel many copies of the system, and selecting the replicas which have reached the highest values along a chosen one-dimensional reaction coordinate. This reaction coordinate does not need to precisely describe all the metastabilities of the system for the method to give reliable results. An extension of the algorithm to compute transition times from one metastable state to another one is also presented.We demonstrate the interest of the method on two simple cases: a onedimensional two-well potential and a two-dimensional potential exhibiting two channels to pass from one metastable state to another one.

Abstract. Let X be a random element in a metric space (F, d), and let Y be a random variable with value 0 or 1. Y is called the class, or the label, of X. Let (Xi, Yi) 1≤i≤n be an observed i.i.d. sample having the same law as (X, Y ). The problem of classification is to predict the label of a new random element X. The k-nearest neighbor classifier is the simple following rule: look at the k nearest neighbors of X in the trial sample and choose 0 or 1 for its label according to the majority vote. Stone (1977) proved in 1977 the universal consistency of this classifier: its probability of error converges to the Bayes error, whatever the distribution of (X, Y ). We show in this paper that this result is no longer valid in general metric spaces. However, if (F, d) is separable and if some regularity condition is assumed, then the k-nearest neighbor classifier is weakly consistent.Mathematics Subject Classification. 62H30.

Abstract. G. Tardos [1] was the first to give a construction of a fingerprinting code whose length meets the lowest known bound in O(c 2 log). This was a real breakthrough because the construction is very simple. Its efficiency comes from its probabilistic nature. However, although G. Tardos almost gave no argument of his rationale, many parameters of his code are precisely fine-tuned. This paper proposes this missing rationale supporting the code construction. The key idea is to render the statistics of the scores as independent as possible from the collusion process. Tardos optimal parameters are rediscovered. This interpretation allows small improvements when some assumptions hold on the collusion process.

We present a novel optical transmission system to experimentally demonstrate the possibility of mode division multiplexing. Its key components are mode multiplexer and demultiplexer based on a programmable liquid crystal on silicon panel, a prototype few-mode fiber, and a 4×4 multiple input multiple output algorithm processing the information of two polarization diversity coherent receivers. Using this system, we transmit two 100 Gb/s PDM-QPSK data streams modulated on two different modes of the prototype few-mode fiber. After 40 km, we obtain Q(2)-factors about 1 dB above the limit for forward error correction.

Let F be a separable Banach space, and let (X, Y ) be a random pair taking values in F × R. Motivated by a broad range of potential applications, we investigate rates of convergence of the k-nearest neighbor estimate r n (x) of the regression function r(x) = E[Y |X = x], based on n independent copies of the pair (X, Y ). Using compact embedding theory, we present explicit and general finite sample bounds on the expected squared difference E[r n (X) − r(X)] 2 , and particularize our results to classical function spaces such as Sobolev spaces, Besov spaces and reproducing kernel Hilbert spaces.