162 research outputs found
Intertwining and commutation relations for birth-death processes
Given a birth-death process on with semigroup
and a discrete gradient depending on a positive weight , we
establish intertwining relations of the form
, where is the Feynman-Kac
semigroup with potential of another birth-death process. We provide
applications when is nonnegative and uniformly bounded from below,
including Lipschitz contraction and Wasserstein curvature, various functional
inequalities, and stochastic orderings. Our analysis is naturally connected to
the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death
processes. The proofs are remarkably simple and rely on interpolation,
commutation, and convexity.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ433 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Confidence regions for the multinomial parameter with small sample size
Consider the observation of n iid realizations of an experiment with d>1
possible outcomes, which corresponds to a single observation of a multinomial
distribution M(n,p) where p is an unknown discrete distribution on {1,...,d}.
In many applications, the construction of a confidence region for p when n is
small is crucial. This concrete challenging problem has a long history. It is
well known that the confidence regions built from asymptotic statistics do not
have good coverage when n is small. On the other hand, most available methods
providing non-asymptotic regions with controlled coverage are limited to the
binomial case d=2. In the present work, we propose a new method valid for any
d>1. This method provides confidence regions with controlled coverage and small
volume, and consists of the inversion of the "covering collection"' associated
with level-sets of the likelihood. The behavior when d/n tends to infinity
remains an interesting open problem beyond the scope of this work.Comment: Accepted for publication in Journal of the American Statistical
Association (JASA
Central limit theorems for additive functionals of ergodic Markov diffusions processes
We revisit functional central limit theorems for additive functionals of
ergodic Markov diffusion processes. Translated in the language of partial
differential equations of evolution, they appear as diffusion limits in the
asymptotic analysis of Fokker-Planck type equations. We focus on the square
integrable framework, and we provide tractable conditions on the infinitesimal
generator, including degenerate or anomalously slow diffusions. We take
advantage on recent developments in the study of the trend to the equilibrium
of ergodic diffusions. We discuss examples and formulate open problems
On gradient bounds for the heat kernel on the Heisenberg group
It is known that the couple formed by the two dimensional Brownian motion and
its L\'evy area leads to the heat kernel on the Heisenberg group, which is one
of the simplest sub-Riemannian space. The associated diffusion operator is
hypoelliptic but not elliptic, which makes difficult the derivation of
functional inequalities for the heat kernel. However, Driver and Melcher and
more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel
on the Heisenberg group. We provide in this paper simple proofs of these
bounds, and explore their consequences in terms of functional inequalities,
including Cheeger and Bobkov type isoperimetric inequalities for the heat
kernel.Comment: Minor correction
Étude spectrale minutieuse de processus moins indécis que les autres
International audienceIn this paper we are looking for quantitative estimates for the convergene to equilibrium of non reversible Markov processes, especialy in short times. The models studied are simple enough to get an explicit expression of the L2 distance betweeen the semigroup and the invariant measure throught time and to compare it with the corresponding reversible cases
Monotonicity of the logarithmic energy for random matrices
It is well-known that the semi-circle law, which is the limiting distribution
in the Wigner theorem, is the minimizer of the logarithmic energy penalized by
the second moment. A very similar fact holds for the Girko and
Marchenko--Pastur theorems. In this work, we shed the light on an intriguing
phenomenon suggesting that this functional is monotonic along the mean
empirical spectral distribution in terms of the matrix dimension. This is
reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann
equation, the monotonicity of the free energy along ergodic Markov processes,
and the Shannon monotonicity of entropy or free entropy along the classical or
free central limit theorem. While we only verify this monotonicity phenomenon
for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square
Laguerre unitary ensemble, numerical simulations suggest that it is actually
more universal. We obtain along the way explicit formulas of the logarithmic
energy of the mentioned models which can be of independent interest
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