Large deviations principle for the Adaptive Multilevel Splitting Algorithm in an idealized setting

The Adaptive Multilevel Splitting (AMS) algorithm is a powerful and versatile method for the simulation of rare events. It is based on an interacting (via a mutation-selection procedure) system of replicas, and depends on two integer parameters: n $\in$ N * the size of the system and the number k $\in$ {1, . . . , n -- 1} of the replicas that are eliminated and resampled at each iteration. In an idealized setting, we analyze the performance of this algorithm in terms of a Large Deviations Principle when n goes to infinity, for the estimation of the (small) probability P(X \textgreater{} a) where a is a given threshold and X is real-valued random variable. The proof uses the technique introduced in [BLR15]: in order to study the log-Laplace transform, we rely on an auxiliary functional equation. Such Large Deviations Principle results are potentially useful to study the algorithm beyond the idealized setting, in particular to compute rare transitions probabilities for complex high-dimensional stochastic processes

Stochastic Models for the 3x+1 and 5x+1 Problems

This paper discusses stochastic models for predicting the long-time behavior of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1 problem. The stochastic models are rigorously analyzable, and yield heuristic predictions (conjectures) for the behavior of 3x+1 orbits and 5x+1 orbits.Comment: 68 pages, 9 figures, 4 table

Successive normalization of rectangular arrays

Standard statistical techniques often require transforming data to have mean $0$ and standard deviation $1$. Typically, this process of "standardization" or "normalization" is applied across subjects when each subject produces a single number. High throughput genomic and financial data often come as rectangular arrays where each coordinate in one direction concerns subjects who might have different status (case or control, say), and each coordinate in the other designates "outcome" for a specific feature, for example, "gene," "polymorphic site" or some aspect of financial profile. It may happen, when analyzing data that arrive as a rectangular array, that one requires BOTH the subjects and the features to be "on the same footing." Thus there may be a need to standardize across rows and columns of the rectangular matrix. There arises the question as to how to achieve this double normalization. We propose and investigate the convergence of what seems to us a natural approach to successive normalization which we learned from our colleague Bradley Efron. We also study the implementation of the method on simulated data and also on data that arose from scientific experimentation.Comment: Published in at http://dx.doi.org/10.1214/09-AOS743 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org). With Correction

Exponential distribution for the occurrence of rare patterns in Gibbsian random fields

We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.Comment: To appear in Commun. Math. Phy