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We consider an estimate of the mode θ of a multivariate probability density f with support in $\mathbb R^d$ using a kernel estimate fn drawn from a sample X1,...,Xn. The estimate θn is defined as any x in {X1,...,Xn} such that $f_n(x)=\max_{i=1, \hdots,n} f_n(X_i)$. It is shown that θn behaves asymptotically as any maximizer ${\hat{\theta}}_n$ of fn. More precisely, we prove that for any sequence $(r_n)_{n\geq 1}$ of positive real numbers such that $r_n\to\infty$ and $r_n^d\log n/n\to 0$, one has $r_n\,\|\theta_n-{\hat{\theta}}_n\| \to 0$ in probability. The asymptotic normality of θn follows without further work.
The paper develops an approach to optimal design problems based on
application of abstract optimisation principles in the space of
measures. Various design criteria and constraints, such as bounded
density, fixed barycentre, fixed variance, etc. are treated in a
unified manner providing a universal variant of the Kiefer-Wolfowitz
theorem and giving a full spectrum of optimality criteria for
particular cases. Incorporating the optimal design problems into
conventional optimisation framework makes it possible to use the
whole arsenal of descent algorithms from the general optimisation
literature for finding optimal designs. The corresponding steepest
descent involves adding a signed measure at every step and converges
faster than the conventional sequential algorithms used to construct
optimal designs. We study a new class of design problems when the
observation points are distributed according to a Poisson point
process arising in the situation when the total control on the
placement of measurements is impossible.
Let Y be a Ornstein–Uhlenbeck diffusion governed by a
stationary and ergodic process X : dYt = a(Xt)Ytdt + σ(Xt)dWt,Y0 = y0.
We establish that under the condition α = Eµ(a(X0)) < 0 with μ the stationary distribution of
the regime process X, the diffusion
Y is ergodic.
We also consider conditions for the
existence of moments for the
invariant law of Y when X is a Markov jump process
having a finite number of states.
Using results on random difference equations
on one hand and the fact that conditionally to
X, Y is Gaussian on the other hand,
we give such a condition for the existence of
the moment of order s ≥ 0. Actually we recover in this case
a result that
Basak et al. [J. Math. Anal. Appl. 202 (1996) 604–622]
have established using the theory of stochastic control of
linear systems.
Using probabilistic tools, this work states a pointwise convergence of
function solutions of the 2-dimensional Boltzmann equation to the function
solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of
Fournier (2000) on the Malliavin calculus for the Boltzmann
equation. Moreover, using the particle system introduced by Guérin and
Méléard (2003), some simulations of the solution of the Landau equation will be given. This result is
original and has not been obtained for the moment by analytical methods.
We consider the continuous time,
one-dimensional random walk in random environment
in Sinai's regime. We show that the probability for the
particle to be, at time t and in a typical environment,
at a distance larger than ta (0<a<1)
from its initial position, is exp{-Const ⋅ ta/[(1 - a)lnt](1 + o(1))}.
A sample of i.i.d. continuous time Markov chains being
defined, the sum over each component of a real function of the
state is considered. For this functional, a central limit theorem
for the first hitting time of a prescribed level is proved.
The result extends the classical central limit theorem for order statistics.
Various reliability models are presented as examples of applications.
We provide an extension of topological methods applied to a
certain class of Non Feller Models which we call Quasi-Feller. We
give conditions to ensure the existence of a stationary
distribution. Finally, we strengthen the conditions to obtain a
positive Harris recurrence, which in turn implies the existence
of a strong law of large numbers.
We present several functional inequalities
for finite difference gradients, such as
a Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities,
associated deviation estimates,
and an exponential integrability property.
In the particular case of the geometric distribution on ${\mathbb{N}}$
we use an integration by parts formula to compute
the optimal isoperimetric and Poincaré constants,
and to obtain an improvement of our
general logarithmic Sobolev inequality.
By a limiting procedure we recover the corresponding
inequalities for the exponential distribution.
These results have applications to interacting spin systems under
a geometric reference measure.
The LISDLG process denoted by J(t) is defined in Iglói and Terdik [ESAIM: PS7 (2003) 23–86] by a
functional limit theorem as the limit of ISDLG processes. This paper gives a
more general limit representation of J(t). It is shown that process J(t)
has its own renormalization group and that J(t) can be represented as the
limit process of the renormalization operator flow applied to the elements of
some set of stochastic processes. The latter set consists of IGSDLG processes
which are generalizations of the ISDLG process.
The stochastic approximation version of EM (SAEM) proposed by Delyon et al. (1999) is
a powerful alternative to EM when the E-step is intractable. Convergence of
SAEM toward a maximum of the observed likelihood is established when
the unobserved data are simulated at each iteration under the conditional
distribution. We show that this very restrictive assumption can be weakened. Indeed,
the results of Benveniste et al. for stochastic approximation
with Markovian perturbations are used to establish the convergence
of SAEM when it is coupled with a Markov chain Monte-Carlo
procedure. This result is very useful for many practical applications.
Applications to the convolution model and the change-points model are presented to illustrate the proposed method.
We consider a diffusion process Xt 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 a
function α of its variance. In order to exhibit global
tests of hypothesis, we derive here central limit theorems for
the Lp deviations such as
\[
\frac1{\sqrt{h}}\left(\frac{h}\varepsilon\right)^{\frac{p}2}\left(
\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p-
\mbox{I E}\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p
\right).
\]
We present a method for estimating the edge of a two-dimensional
bounded set, given a finite random set of points drawn from the interior.
The estimator is based both on a Parzen-Rosenblatt kernel and
extreme values of point processes. We give conditions
for various kinds of convergence and asymptotic normality.
We propose a method of reducing the negative bias and edge effects,
illustrated by some simulations.
The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on $\mathbb{Z}^d$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster.
As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation.
We also prove a flat edge result in the case of dimension 2. Various examples are also given.
We study the large deviation principle for stochastic processes of the form $\{\sum_{k=1}^{\infty}x_{k}(t)\xi_{k}:t\in T\}$, where $\{\xi_{k}\}_{k=1}^{\infty}$ is a sequence of i.i.d.r.v.'s with mean zero and $x_{k}(t)\in \mathbb{R}$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition, we derive new concentration inequalities, which are of independent interest.