32 research outputs found
Poisson-type deviation inequalities for curved continuous-time Markov chains
In this paper, we present new Poisson-type deviation inequalities for
continuous-time Markov chains whose Wasserstein curvature or -curvature
is bounded below. Although these two curvatures are equivalent for Brownian
motion on Riemannian manifolds, they are not comparable in discrete settings
and yield different deviation bounds. In the case of birth--death processes, we
provide some conditions on the transition rates of the associated generator for
such curvatures to be bounded below and we extend the deviation inequalities
established [An\'{e}, C. and Ledoux, M. On logarithmic Sobolev inequalities for
continuous time random walks on graphs. Probab. Theory Related Fields 116
(2000) 573--602] for continuous-time random walks, seen as models in null
curvature. Some applications of these tail estimates are given for
Brownian-driven Ornstein--Uhlenbeck processes and queues.Comment: Published at http://dx.doi.org/10.3150/07-BEJ6039 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature
The purpose of this paper is to extend the investigation of Poisson-type
deviation inequalities started by Joulin (Bernoulli 13 (2007) 782--798) to the
empirical mean of positively curved Markov jump processes. In particular, our
main result generalizes the tail estimates given by Lezaud (Ann. Appl. Probab.
8 (1998) 849--867, ESAIM Probab. Statist. 5 (2001) 183--201). An application to
birth--death processes completes this work.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ158 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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
Measure concentration through non-Lipschitz observables and functional inequalities
Non-Gaussian concentration estimates are obtained for invariant probability
measures of reversible Markov processes. We show that the functional
inequalities approach combined with a suitable Lyapunov condition allows us to
circumvent the classical Lipschitz assumption of the observables. Our method is
general and covers diffusions as well as pure-jump Markov processes on
unbounded spaces
Curvature, concentration and error estimates for Markov chain Monte Carlo
We provide explicit nonasymptotic estimates for the rate of convergence of
empirical means of Markov chains, together with a Gaussian or exponential
control on the deviations of empirical means. These estimates hold under a
"positive curvature" assumption expressing a kind of metric ergodicity, which
generalizes the Ricci curvature from differential geometry and, on finite
graphs, amounts to contraction under path coupling.Comment: Published in at http://dx.doi.org/10.1214/10-AOP541 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
INTERTWININGS AND GENERALIZED BRASCAMP-LIEB INEQUALITIES
International audienceWe continue our investigation of the intertwining relations for Markov semigroups and extend the results of [9] to multi-dimensional diffusions. In particular these formulae entail new functional inequalities of Brascamp-Lieb type for log-concave distributions and beyond. Our results are illustrated by some classical and less classical examples
A note on spectral gap and weighted Poincar\'e inequalities for some one-dimensional diffusions
International audienceWe present some classical and weighted Poincar\'e inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors for a class of spherically symmetric probability measures in dimension larger than 2. Our strategy is based on two main ingredients: on the one hand, the optimal constant in the desired weighted Poincar\'e inequality has to be rewritten as the spectral gap of a convenient Markovian diffusion operator, and on the other hand we use a recent result given by the two first authors, which allows to estimate precisely this spectral gap. In particular we are able to capture its exact value for some examples
Upper bounds on Rubinstein distances on configuration spaces and applications
In this paper, we provide upper bounds on several Rubinstein-type distances
on the configuration space equipped with the Poisson measure. Our inequalities
involve the two well-known gradients, in the sense of Malliavin calculus, which
can be defined on this space. Actually, we show that depending on the distance
between configurations which is considered, it is one gradient or the other
which is the most effective. Some applications to distance estimates between
Poisson and other more sophisticated processes are also provided, and an
application of our results to tail and isoperimetric estimates completes this
work.Comment: To appear in Communications on Stochastic Analysis and Application