The transition kernel of the well-known Metropolis-Hastings (MH) algorithm has a point mass at the chain’s current position, which prevent direct smoothness properties to be derived for the successive densities of marginals issued from this algorithm. We show here that under mild smoothness assumption on the MH algorithm “input” densities (the initial, proposal and target distributions), propagation of a Lipschitz condition for the iterative densities can be proved. This allows us to build a consistent nonparametric estimate of the entropy for these iterative densities. This theoretical study can be viewed as a building block for a more general MCMC evaluation tool grounded on such estimates.

In Benaïm and Ben Arous (2003) is solved a multi-armed bandit problem arising in the theory of learning in games. We proposea short and elementary proof of this result based on a variant of the Kronecker lemma.

We study the numerical approximation of doubly reflected backward stochastic differentialequations with intermittent upper barrier (RIBSDEs). These denote reflected BSDEs in whichthe upper barrier is only active on certain random time intervals. From the point of viewof financial interpretation, RIBSDEs arise as pricing equations of game options withconstrained callability. In a Markovian set-up we prove a convergence rate for atime-discretization scheme by simulation to an RIBSDE. We also characterize the solutionof an RIBSDE as the largest viscosity subsolution of a related system of variationalinequalities, and we establish the convergence of a deterministic numerical scheme forthat problem. Due to the potentially very high dimension of the system of variationalinequalities, this approach is not always practical. We thus subsequently prove aconvergence rate for a time-discretisation scheme by simulation to an RIBSDE.

In the present paper we prove moderate deviations for a Curie–Weiss model with external magnetic field generated by a dynamical system, as introduced by Dombry and Guillotin-Plantard in [C. Dombry and N. Guillotin-Plantard, Markov Process. Related Fields 15 (2009) 1–30]. The results extend those already obtained for the Curie–Weiss model without external field by Eichelsbacher and Löwe in [P. Eichelsbacher and M. Löwe, Markov Process. Related Fields 10 (2004) 345–366]. The Curie–Weiss model with dynamical external field is related to the so called dynamic ℤ-random walks (see [N. Guillotin-Plantard and R. Schott, Theory and applications, Elsevier B. V., Amsterdam (2006).]). We also prove a moderate deviation result for the dynamic ℤ-random walk, completing the list of limit theorems for this object.

We study the stochastic homogenization processes considered by Baldi (1988) and by Facchinetti and Russo (1983). We precise the speed of convergence towards the homogenized state by proving the following results: (i) a large deviations principle holds for the Young measures; if the Young measures are evaluated on a given function, then (ii) the speed of convergence is bounded in every Lp norm by an explicit rate and (iii) central limit theorems hold. In dimension 1, we apply these results to the stochastic homogenization of random p-Laplacian operators for any p > 1.

The dual attainment of the Monge–Kantorovich transport problem is analyzed in a generalsetting. The spaces X,Y are assumed to be polish and equipped with Borelprobability measures μ and ν. The transport costfunction c : X × Y →  [0,∞]  is assumedto be Borel measurable. We show that a dual optimizer always exists, provided we interpretit as a projective limit of certain finitely additive measures. Our methods are functionalanalytic and rely on Fenchel’s perturbation technique.

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}\sum_1^n\mathbf{f}(x_i^n) \cdot Z^n_i$ where the deterministic probability measure $\frac{1}{n}\sum_1^n \delta_{x_i^n}$ converges weakly to a probability measure R and $(Z^n_i)_{i\in \mathbb{N}}$ are $\mathbb{R}^d$ -valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition: $$ \sup_{i,n} {\mathbb E}{\rm e}^{\alpha^* |Z_i^n|}< +\infty \quad\textrm{for some}\quad 0<\alpha^*<+\infty.$$

In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend Erdös and Rényi's functional law of large numbers.

The sequence of random probability measures ν n that gives a path of length n, $\frac{1}{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment.Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.

This paper is devoted to the study of some asymptotic properties of aM-estimator in a framework of detection of abrupt changes inrandom field's distribution. This class of problems includes e.g.recovery of sets. It involves various techniques, including M-estimation method, concentrationinequalities, maximal inequalities for dependent random variables and ϕ-mixing. Penalization of the criterion function when the size of thetrue model is unknown is performed. All the results apply under mild, discussedassumptions. Simple examples are provided.

The paper is concerned with the asymptotic distributions of estimators for the length and the centre of the so-called η-shorth interval in a nonparametric regression framework. It is shown that the estimator of thelength converges at the n 1/2-rate to a Gaussian law and that the estimator of the centre converges at the n 1/3-rate to the locationof the maximum of a Brownian motion with parabolic drift.Bootstrap procedures are proposed and shown to be consistent. They are compared with the plug-in method through simulations.

A discrete time stochastic model for a multiagent system given in terms of a large collection of interacting Markov chains is studied. The evolution of the interacting particles is described through a time inhomogeneous transition probability kernel that depends on the ‘gradient’ of the potential field. The particles, in turn, dynamically modify the potential field through their cumulative input. Interacting Markov processes of the above form have been suggested as models for active biological transport in response to external stimulus such as a chemical gradient. One of the basic mathematical challenges is to develop a general theory of stability for such interacting Markovian systems and for the corresponding nonlinear Markov processes that arise in the large agent limit. Such a theory would be key to a mathematical understanding of the interactive structure formation that results from the complex feedback between the agents and the potential field. It will also be a crucial ingredient in developing simulation schemes that are faithful to the underlying model over long periods of time. The goal of this work is to study qualitative properties of the above stochastic system as the number of particles (N) and the time parameter (n) approach infinity. In this regard asymptotic properties of a deterministic nonlinear dynamical system, that arises in the propagation of chaos limit of the stochastic model, play a key role. We show that under suitable conditions this dynamical system has a unique fixed point. This result allows us to study stability properties of the underlying stochastic model. We show that as N → ∞, the stochastic system is well approximated by the dynamical system, uniformly over time. As a consequence, for an arbitrarily initialized system, as N → ∞ and n → ∞, the potential field and the empirical measure of the interacting particles are shown to converge to the unique fixed point of the dynamical system. In general, simulation of such interacting Markovian systems is a computationally daunting task. We propose a particle based approximation for the dynamic potential field which allows for a numerically tractable simulation scheme. It is shown that this simulation scheme well approximates the true physical system, uniformly over an infinite time horizon.

We consider the high order moments estimator of the frontier of a random pair, introduced by [S. Girard, A. Guillou and G. Stupfler, J. Multivariate Anal. 116 (2013) 172–189]. In the present paper, we show that this estimator is strongly uniformly consistent on compact sets and its rate of convergence is given when the conditional cumulative distribution function belongs to the Hall class of distribution functions.

We prove the existence and smoothness of density for the solutionof a hyperbolic SPDE with free term coefficients depending ontime, under hypoelliptic non degeneracy conditions. The resultextends those proved in Cattiaux and Mesnager, PTRF123 (2002) 453-483 to an infinite dimensionalsetting.

We consider a diffusion process X t smoothed with (small)sampling parameter ε. As in Berzin, León and Ortega(2001), we consider a kernel estimate $\widehat{\alpha}_{\varepsilon}$ with window h(ε) of afunction α of its variance. In order to exhibit globaltests of hypothesis, we derive here central limit theorems forthe Lp deviations such as \[ \frac1{\sqrt{h}}\left(\frac{h}\varepsilon\right)^{\frac{p}2}\left(\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p-\mbox{I E}\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p\right).\]

In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increasedfrequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is in fact typical for “rare events” defined as long cylinder sets in processes generated by a finite partition of an arbitrary (infinite aperiodic) ergodic measure preserving transformation.

We consider an extension of the Kyle and Back's model [Back, Rev. Finance Stud.5 (1992) 387–409; Kyle, Econometrica35 (1985) 1315–1335],meaning a model for the market with a continuous time risky assetand asymmetrical information. There are threefinancial agents: the market maker, an insider trader (who knows a randomvariable V which will be revealed at final time) and a non informed agent. Here we assume thatthe non informed agent is strategic, namely he/she uses a utility function to optimize his/her strategy.Optimal control theory is applied to obtain a pricing ruleand to prove the existenceof an equilibrium price when the insider trader and the non informedagent are risk-neutral. We will show that if such an equilibrium exists, then the non informed agent's optimal strategy is to do nothing, in other words to be non strategic.

We establish necessary and sufficient conditions for the convergence (in the sense offinite dimensional distributions) of multiplicative measures on the set of partitions. Themultiplicative measures depict distributions of component spectra of random structures andalso the equilibria of classic models of statistical mechanics and stochastic processes ofcoagulation-fragmentation. We show that the convergence of multiplicative measures isequivalent to the asymptotic independence of counts of components of fixed sizes in randomstructures. We then apply Schur’s tauberian lemma and some results from additive numbertheory and enumerative combinatorics in order to derive plausible sufficient conditions ofconvergence. Our results demonstrate that the common belief, that counts of components offixed sizes in random structures become independent as the number of particles goes toinfinity, is not true in general.

We study higher-order moment measures of heavy-tailed renewal models, including a renewalpoint process with heavy-tailed inter-renewal distribution and its continuous analog, theoccupation measure of a heavy-tailed Lévy subordinator. Our results reveal that theasymptotic structure of such moment measures are given by explicit power-law densityfunctions. The same power-law densities appear naturally as cumulant measures of certainPoisson and Gaussian stochastic integrals. This correspondence provides new and extendedresults regarding the asymptotic fluctuations of heavy-tailed sources under aggregation,and clarifies existing links between renewal models and fractional random processes.

The Lévy transform of a Brownian motion B is the Brownian motionB(1) given by Bt(1) = ∫0tsgn(Bs)dBs; callB(n) the Brownian motion obtained fromB by iterating n times this transformation. Weestablish that almost surely, the sequence of paths (t → Bt(n))n⩾0isdense in Wiener space, for the topology of uniform convergence on compact timeintervals.

Equip the edges of the lattice ℤ2 with i.i.d. random capacities. A law oflarge numbers is known for the maximal flow crossing a rectangle in ℝ2 when theside lengths of the rectangle go to infinity. We prove that the lower large deviations areof surface order, and we prove the corresponding large deviation principle from below.This extends and improves previous large deviations results of Grimmett and Kesten [9] obtained for boxes of particular orientation.