The paper is motivated by the stochastic comparison of the reliabilityof non-repairable k-out-of-n systems.The lifetime of such a system with nonidentical components is compared with the lifetime of a system withidentical components.Formally the problem is as follows. Let Ui,i = 1,...,n, be positiveindependent random variables with common distribution F.For λi > 0 and µ > 0, let considerXi = Ui/λi and Yi = Ui/µ, i = 1,...,n. Remark that this is no more than a change of scale for each term.For k ∈ {1,2,...,n}, let us define Xk:n to be the kth order statistics of the random variables X1,...,Xn , andsimilarly Yk:n to be the kth order statistics ofY1,...,Yn .If Xi,i = 1,...,n, are the lifetimes of the components of an+1-k-out-of-n non-repairable system, then Xk:n is thelifetime of the system.In this paper, we give for a fixed k a sufficient condition forXk:n ≥st Yk:n where st is the usual ordering for distributions.In the Markovian case (all components have an exponential lifetime), wegive a necessary and sufficient condition.We prove that Xk:n is greater that Yk:n according to the usualstochastic ordering if and only if \[\left( \begin{array}{c} n k \end{array}\right){\mu}^k\geq \sum_{1\leqi_1<i_2<...<i_k\leq n}\lambda_{i_1}\lambda_{i_2}...\lambda_{i_k}.\]

In many practical situations sample sizes are not sufficiently largeand estimators based on such samples may not be satisfactory interms of their variances. At the same time it is not unusual thatsome auxiliary information about the parameters of interest isavailable. This paper considers a method of using auxiliaryinformation for improving properties of the estimators based on acurrent sample only. In particular, it is assumed that theinformation is available as a number of estimates based on samplesobtained from some other mutually independent data sources. Thismethod uses the fact that there is a correlation effect betweenestimators based on the current sample and auxiliary informationfrom other sources. If variance covariance matrices of vectors ofestimators used in the estimating procedure are known, this methodproduces more efficient estimates in terms of their variancescompared to the estimates based on the current sample only. If thesevariance-covariance matrices are not known, their consistentestimates can be used as well such that the large sample propertiesof the method remain unchangeable. This approach allows to improvestatistical properties of many standard estimators such as anempirical cumulative distribution function, empirical characteristicfunction, and Nelson-Aalen cumulative hazard estimator.

Given an n-sample from some unknown density f on [0,1], it is easy to construct anhistogram of the data based on some given partition of [0,1], but not so much is knownabout an optimal choice of the partition, especially when the data set is not large, even ifone restricts to partitions into intervals of equal length. Existing methods are either rulesof thumbs or based on asymptotic considerations and often involve some smoothnessproperties of f. Our purpose in this paper is to give an automatic, easy to program andefficient method to choose the number of bins of the partition from the data. It is based on boundson the risk of penalized maximum likelihood estimators due to Castellan and heavy simulationswhich allowed us to optimize the form of the penalty function. These simulations show that themethod works quite well for sample sizes as small as 25.

Initially motivated by a practical issue in target detection vialaser vibrometry, we are interested in the problem of periodicsignal detection in a Gaussian fixed design regression framework.Assuming that the signal belongs to some periodic Sobolev ball andthat the variance of the noise is known, we first consider theproblem from a minimax point of view: we evaluate the so-calledminimax separation rate which corresponds to the minimall 2-distance between the signal and zero so that the detection ispossible with prescribed probabilities of error. Then, we propose atesting procedure which is available when the variance of the noiseis unknown and which does not use any prior information about thesmoothness degree or the period of the signal. We prove that it isadaptive in the sense that it achieves, up to a possible logarithmicfactor, the minimax separation rate over various periodic Sobolevballs simultaneously. The originality of our approach as compared torelated works on the topic of signal detection is that our testingprocedure is sensitive to the periodicity assumption on the signal.A simulation study is performed in order to evaluate the effect ofthis prior assumption on the power of the test. We do observe thegains that we could expect from the theory. At last, we turn to theapplication to target detection by laser vibrometry that we had inview.

Soit U une fonction définie sur un ensemble fini E muni d'unnoyau markovien irréductible M. L'objectif du papier est de comparerthéoriquement deux procédures stochastiques de minimisation globale de U :le recuit simulé et un algorithme génétique.Pour ceci on se placera dans la situation idéalisée d'une infinité de particules disponibleset nous feronsune hypothèse commode d'existence de suffisamment de symétries du cadre (E,M,U).On verra notamment que contrairement au recuit simulé, toute évolution logarithmiquede l'inverse de la température conduit asymptotiquement à concentrer les dynamiques de Feynman-Kac recuitessurcertains minima globaux de U (du moins si M ne fait pas sortir de leur ensemble en un temps presque immédiat).Par contre, ces dernières ont tendance à oublier plus difficilement leur condition initiale et on quantifiera précisément cette propriété. Enfin on présentera les conjectures correspondantes dans une situation plus générale.

In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surelyfinite random terminal time. More precisely, we are going to show that if (Wn ) is a sequence of scaled random walks or a sequence of martingales that converges to a Brownian motion W and if $(\tau^n)$ is a sequence of stopping times that converges to a stopping time τ, then the solution of the BSDE driven by Wn with random terminal time $\tau^n$ converges to the solution of the BSDE driven by W with random terminal time τ.

We propose a method based on a penalised likelihood criterion, forestimating the number on non-zero components of the meanof a Gaussian vector. Following the work of Birgé and Massart in Gaussian modelselection, we choose the penalty function such that the resultingestimator minimises the Kullback risk.

We consider a system of fully coupled forward-backward stochasticdifferential equations. First we generalize the results ofPardoux-Tang [7] concerning the regularity of the solutions withrespect to initial conditions. Then, we prove that in some particularcases this system leads to aprobabilistic representation of solutions of a second-order PDE whosesecond order coefficients depend on the gradient of the solution. Wethen give some examples in dimension 1 and dimension 2 for which theassumptions are easy to check.

We extend Hoggar's theorem that the sum of two independentdiscrete-valued log-concave random variables is itself log-concave. Weintroduce conditions under which the result still holds for dependentvariables. We argue that these conditions are natural by giving someapplications. Firstly, we use our main theorem to give simple proofsof the log-concavity of the Stirling numbers of the second kind and ofthe Eulerian numbers.Secondly, we prove results concerning the log-concavityof the sum of independent (not necessarily log-concave) randomvariables.

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 branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independent of random motion, and intensities of reverses are defined by a particle's current direction. A solution of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) has a so-called McKean representation via such processes. Commonly this system possesses travelling-wave solutions. The convergence of solutions with Heaviside terminal data to the travelling waves is discussed. The paper realizes the McKean's program for the Kolmogorov-Petrovskii-Piskunov equation in this case. The Feynman-Kac formula plays a key role.

For a sequence of blockwise m-dependent random variables {Xn,n ≥ 1}, conditions are provided under which $\lim_{n\to\infty}(\sum_{i=1}^nX_i)/b_n=0$ almost surely where {bn,n ≥ 1} is a sequence of positive constants. The results are new even when bn ≡ nr,r > 0. As special case, the Brunk-Chung strong law of large numbers is obtained for sequences of independent random variables. The current work also extends results of Móricz [Proc. Amer. Math. Soc.101 (1987) 709–715], and Gaposhkin [Teor. Veroyatnost. i Primenen. 39 (1994) 804–812]. The sharpness of the results is illustrated by examples.

Unbiased risk estimation, à la Stein, is studied for infinitely divisible laws with finite second moment.

We study the probability distribution of the location of a particleperforming a cyclic random motion in $\mathbb{R}^d$ . The particle can taken possible directions with different velocities and the changes ofdirection occur at random times. The speed-vectors as well as thesupport of the distribution form a polyhedron (the first one havingconstant sides and the other expanding with time t). Thedistribution of the location of the particle is made up of twocomponents: a singular component (corresponding to the beginning ofthe travel of the particle) and an absolutely continuous component.
We completely describe the singular component and exhibit anintegral representation for the absolutely continuous one. Thedistribution is obtained by using a suitable expression of thelocation of the particle as well as some probability calculustogether with some linear algebra. The particular case of theminimal cyclic motion (n=d+1) with Erlangian switching times isalso investigated and the related distribution can be expressed interms of hyper-Bessel functions with several arguments.

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/∞ queues.Proofs are elementary and rely essentially on the development of a “Φ-calculus”.

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 iscalled 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 topredict the label of a new random element X. The k-nearestneighbor classifier is the simple following rule: look atthe k nearest neighbors of X in the trial sample and choose 0 or 1 for its labelaccording to the majority vote. When $({\cal F},d)=(\mathbb{R}^d,||.||)$ ,Stone (1977) proved in 1977 the universal consistency of this classifier:its probability of error converges to the Bayes error, whatever thedistribution of (X,Y). We show in this paper that this result is nolonger valid in general metric spaces. However, if (F,d) isseparable and if some regularity condition is assumed, then thek-nearest neighbor classifier is weakly consistent.

We show in this article how the theory of “rough paths”allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.

We study strictly parabolic stochastic partial differential equations on $\mathbb{R}^d$ , d ≥ 1,driven by a Gaussian noise white in time and coloured in space. Assuming that thecoefficients of the differential operator are random, we give sufficient conditions on thecorrelation of the noise ensuring Hölder continuity for the trajectories of thesolution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weakformulation of the equation are related, deriving path properties of the solution to aparabolic Cauchy problem in evolution form.