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The growth exponent α for loop-erased or Laplacian random walk
on the integer lattice is defined by saying that the expected time to
reach the sphere of radius n is of order nα. We prove that
in two dimensions, the growth exponent is strictly greater than one.
The proof uses a known estimate on the third moment of the escape
probability and an improvement on the discrete Beurling projection theorem.
Minimax bounds for the risk function of estimators of functionals of
the spectral density of Gaussian
fields 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 are
given for all dimensions. For non-linear integral functionals, these
estimators are constructed for the two and three dimensional problems.
Consider a system of many components with constant failure rate and
general repair rate. When all components are reliable and easily reparable,
the reliability of the system can be evaluated from the probability q of
failure before restoration. In , authors give an asymptotic
approximation 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 attractors
of the empirical measure under mean field Gibbs measures.
We prove that a mild change of the densities
of these measures does not affect the central limit theorems.
We apply this result to generalize the assumptions
of  and  on the densities of the Gibbs measures to
get precise Laplace estimates.
We study the convergence to equilibrium of n-samples of independent Markov
chains 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 the
sense that as n tends to infinity, the total variation distance
between the distribution of the chain and the asymptotic distribution tends
to 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  to give bounds to
the distribution of the maximum of a smooth stationary Gaussian
process. We give simpler expressions of the first two terms of
the Rice series [3,13] for the distribution of the maximum.
Our main contribution is a simpler form of the second factorial moment
of the number of upcrossings which is in some sense a generalization
of Steinberg et al.'s formula
( p. 212).
Then, we present a numerical application and asymptotic expansions
that give a new interpretation of a result by
Stein's method is used to prove approximations in total variation to the
distributions 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 earlier
work of Kruopis (1983) and Čekanavičius (1997);
coupling methods are used to derive concrete expressions for the error
bounds. An example is given to illustrate the potential for application
to sums of dependent random variables.