The growth exponent α for loop-erased or Laplacian random walkon the integer lattice is defined by saying that the expected time toreach the sphere of radius n is of order nα . We prove thatin two dimensions, the growth exponent is strictly greater than one.The proof uses a known estimate on the third moment of the escapeprobability and an improvement on the discrete Beurling projection theorem.

Minimax bounds for the risk function of estimators of functionals ofthe spectral density of Gaussianfields are obtained. This result is a generalization of a previous result of Khas'minskii and Ibragimov on Gaussian processes.Efficient estimators are then constructed for these functionals. In the case of linear functionals these estimators aregiven for all dimensions. For non-linear integral functionals, theseestimators are constructed for the two and three dimensional problems.

Consider a system of many components with constant failure rate andgeneral repair rate. When all components are reliable and easily reparable,the reliability of the system can be evaluated from the probability q offailure before restoration. In [14], authors give an asymptoticapproximation by monotone sequences. In the same framework, we propose,here, a bounding for q and apply it in the ageing property case.

We study the fluctuations around non degenerate attractorsof the empirical measure under mean field Gibbs measures.We prove that a mild change of the densitiesof these measures does not affect the central limit theorems.We apply this result to generalize the assumptionsof [3] and [12] on the densities of the Gibbs measures toget precise Laplace estimates.

We study the convergence to equilibrium of n-samples of independent Markovchains in discrete and continuous time. They are defined as Markov chains on the n-fold Cartesian product of the initial state space by itself, and they converge to the direct product of n copies of the initial stationary distribution. Sharp estimates for the convergence speed are given in terms of the spectrum of the initial chain. A cutoff phenomenon occurs in thesense that as n tends to infinity, the total variation distancebetween the distribution of the chain and the asymptotic distribution tendsto 1 or 0 at all times. As an application,an algorithm is proposed for producing an n-sample of the asymptotic distribution of the initial chain, with an explicit stopping test.

This paper uses the Rice method [18] to give bounds tothe distribution of the maximum of a smooth stationary Gaussianprocess. We give simpler expressions of the first two terms ofthe Rice series [3,13] for the distribution of the maximum. Our main contribution is a simpler form of the second factorial momentof the number of upcrossings which is in some sense a generalizationof Steinberg et al.'s formula ([7] p. 212). Then, we present a numerical application and asymptotic expansionsthat give a new interpretation of a result by Piterbarg [15].

Stein's method is used to prove approximations in total variation to thedistributions of integer valued random variables by (possibly signed)compound Poisson measures. For sums of independent random variables,the results obtained are very explicit, and improve upon earlierwork of Kruopis (1983) and Čekanavičius (1997);coupling methods are used to derive concrete expressions for the errorbounds. An example is given to illustrate the potential for applicationto sums of dependent random variables.

We prove a new large deviation inequality with applicationswhen projecting a density on a wavelet basis.