Smoothing equations for large P\'olya urns

Consider a balanced non triangular two-color P\'olya-Eggenberger urn process, assumed to be large which means that the ratio sigma of the replacement matrix eigenvalues satisfies 1/2<sigma <1. The composition vector of both discrete time and continuous time models admits a drift which is carried by the principal direction of the replacement matrix. In the second principal direction, this random vector admits also an almost sure asymptotics and a real-valued limit random variable arises, named WDT in discrete time and WCT in continous time. The paper deals with the distributions of both W. Appearing as martingale limits, known to be nonnormal, these laws remain up to now rather mysterious. Exploiting the underlying tree structure of the urn process, we show that WDT and WCT are the unique solutions of two distributional systems in some suitable spaces of integrable probability measures. These systems are natural extensions of distributional equations that already appeared in famous algorithmical problems like Quicksort analysis. Existence and unicity of the solutions of the systems are obtained by means of contracting smoothing transforms. Via the equation systems, we find upperbounds for the moments of WDT and WCT and we show that the laws of WDT and WCT are moment-determined. We also prove that WDT is supported by the whole real line and admits a continuous density (WCT was already known to have a density, infinitely differentiable on R\{0} and not bounded at the origin)

The relation between tree size complexity and probability for Boolean functions generated by uniform random trees

We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives And and OR and the leaves are labelled with variables or negated variables. We study limiting distribution when the tree size tends to infinity and derive a relation between the tree size complexity and the probability of a function. This is done by first expressing trees representing a particular function as expansions of minimal trees representing this function and then computing the probabilities by means of combinatorial counting arguments relying on generating functions and singularity analysis

Stochastic approximation on non-compact measure spaces and application to measure-valued Pólya processes

Our main result is to prove almost-sure convergence of a stochastic-approximation algorithm defined on the space of measures on a non-compact space. Our motivation is to apply this result to measure-valued P\'olya processes (MVPPs, also known as infinitely-many P\'olya urns). Our main idea is to use Foster-Lyapunov type criteria in a novel way to generalize stochastic-approximation methods to measure-valued Markov processes with a non-compact underlying space, overcoming in a fairly general context one of the major difficulties of existing studies on this subject. From the MVPPs point of view, our result implies almost-sure convergence of a large class of MVPPs, this convergence was only obtained until now for specific examples, with only convergence in probability established for general classes. Furthermore, our approach allows us to extend the definition of MVPPs by adding "weights" to the different colors of the infinitely-many-color urn. We also exhibit a link between non-"balanced" MVPPs and quasi-stationary distributions of Markovian processes, which allows us to treat, for the first time in the literature, the non-balanced case. Finally, we show how our result can be applied to designing stochastic-approximation algorithms for the approximation of quasi-stationary distributions of discrete- and continuous-time Markov processes on non-compact spaces

Equatorial mountain torques and cold surge preconditioning

International audienceThe evolution of the two components of the equatorial mountain torque (EMT) applied by mountains on the atmosphere is analyzed in the NCEP reanalysis. A strong lagged relationship between the EMT component along the Greenwich axis TM1 and the EMT component along the 90°E axis TM2 is found, with a pronounced signal on TM1 followed by a signal of opposite sign on TM2. It is shown that this result holds for themajor massifs (Antarctica, the Tibetan Plateau, the Rockies, and the Andes) if a suitable axis systemis used for each of them. For the midlatitude mountains, this relationship is in part associated with the development of cold surges. Following these results, two hypotheses are made: (i) the mountain forcing on the atmosphere is well measured by the regional EMTs and (ii) this forcing partly drives the cold surges. To support these, a purely dynamical linear model is proposed: it is written on the sphere, uses an f-plane quasigeostrophic approximation, and includes the mountain forcings. In this model, a positive (negative) peak in TM1 produced by a mountain massif in the Northern (Southern) Hemisphere is due to a large-scale high surface pressure anomaly poleward of the massif. At a later stage, high pressure and low temperature anomalies develop in the lower troposphere east of the mountain, explaining the signal on TM2 and providing the favorable conditions for the cold surge development. It is concluded that the EMT is a good measure of the dynamical forcing of the atmospheric flow by the mountains and that the poleward forces exerted by mountains on the atmosphere are substantial drivers of the cold surges, at least in their early stage. Therefore, the EMT time series can be an important diagnostic to assess the representation of mountains in general circulation models. © 2010 American Meteorological Society