380 research outputs found
Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains
We study properties of the Laplace transforms of non-negative additive
functionals of Markov chains. We are namely interested in a multiplicative
ergodicity property used in [18] to study bifurcating processes with ancestral
dependence. We develop a general approach based on the use of the operator
perturbation method. We apply our general results to two examples of Markov
chains, including a linear autoregressive model. In these two examples the
operator-type assumptions reduce to some expected finite moment conditions on
the functional (no exponential moment conditions are assumed in this work)
Additional material on bounds of -spectral gap for discrete Markov chains with band transition matrices
We analyse the -convergence rate of irreducible and aperiodic
Markov chains with -band transition probability matrix and with
invariant distribution . This analysis is heavily based on: first the
study of the essential spectral radius of
derived from Hennion's quasi-compactness criteria; second
the connection between the spectral gap property (SG) of on
and the -geometric ergodicity of . Specifically, (SG)
is shown to hold under the condition \alpha\_0 := \sum\_{{m}=-N}^N
\limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\,
1. Moreover . Simple conditions
on asymptotic properties of and of its invariant probability distribution
to ensure that \alpha\_0\textless{}1 are given. In particular this
allows us to obtain estimates of the -geometric convergence rate
of random walks with bounded increments. The specific case of reversible is
also addressed. Numerical bounds on the convergence rate can be provided via a
truncation procedure. This is illustrated on the Metropolis-Hastings algorithm
The Nagaev-Guivarc'h method via the Keller-Liverani theorem
The Nagaev-Guivarc'h method, via the perturbation operator theorem of Keller
and Liverani, has been exploited in recent papers to establish local limit and
Berry-Essen type theorems for unbounded functionals of strongly ergodic Markov
chains. The main difficulty of this approach is to prove Taylor expansions for
the dominating eigenvalue of the Fourier kernels. This paper outlines this
method and extends it by proving a multi-dimensional local limit theorem, a
first-order Edgeworth expansion, and a multi-dimensional Berry-Esseen type
theorem in the sense of Prohorov metric. When applied to uniformly or
geometrically ergodic chains and to iterative Lipschitz models, the above cited
limit theorems hold under moment conditions similar, or close, to those of the
i.i.d. case
A Renewal Theorem for Strongly Ergodic Markov Chains in Dimension and Centered Case
In dimension , we present a general assumption under which the
renewal theorem established by Spitzer for i.i.d. sequences of centered
nonlattice r.v. holds true. Next we appeal to an operator-type procedure to
investigate the Markov case. Such a spectral approach has been already
developed by Babillot, but the weak perturbation theorem of Keller and Liverani
enables us to greatly weaken thehypotheses in terms of moment conditions. Our
applications concern the v\rho$-mixing Markov chains, and the iterative Lipschitz models, for which the
renewal theorem of the i.i.d. case extends under the (almost) expected moment
condition
A uniform Berry--Esseen theorem on -estimators for geometrically ergodic Markov chains
Let be a -geometrically ergodic Markov chain. Given some
real-valued functional , define
,
. Consider an estimator
, that is, a measurable function of the observations satisfying
with
some sequence of real numbers going to zero. Under some
standard regularity and moment assumptions, close to those of the i.i.d. case,
the estimator satisfies a Berry--Esseen theorem uniformly with
respect to the underlying probability distribution of the Markov chain.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ347 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Regular perturbation of V -geometrically ergodic Markov chains
In this paper, new conditions for the stability of V-geometrically ergodic
Markov chains are introduced. The results are based on an extension of the
standard perturbation theory formulated by Keller and Liverani. The continuity
and higher regularity properties are investigated. As an illustration, an
asymptotic expansion of the invariant probability measure for an autoregressive
model with i.i.d. noises (with a non-standard probability density function) is
obtained
VITESSE DE CONVERGENCE DANS LE THÉORÈME LIMITE CENTRAL POUR CHAÎNES DE MARKOV DE PROBABILITÉ DE TRANSITION QUASI-COMPACTE
NOMBRE DE PAGES : 16International audienceLet be a transition probability on a measurable space , let be a Markov chain associated to , and let be a real-valued measurable function on , and . Under functional hypotheses on the action of and its Fourier kernels , we investigate the rate of convergence in the central limit theorem for the sequence . According to the hypotheses, we prove that the rate is, either for all , or . We apply the spectral method of Nagaev which is improved by using a perturbation theorem of Keller and Liverani and a method of martingale difference reduction. When is not compact or is not bounded, the conditions required here are weaker than the ones usually imposed when the standard perturbation theorem is used. For example, in the case of -geometric ergodic chains or Lipschitz iterative models, the rate of convergence in the c.l.t is under a third moment condition on
Limit theorems for geometrically ergodic Markov chains
17 pagesLet (E,\cE) be a countably generated state space, let be an aperiodic and -irreducible -geometrically ergodic Markov chain on , with and a -finite positive measure on . Let be the -invariant distribution, and let measurable and dominated by . Then \sigma^2 = \lim_n n^{-1}\E_x[(S_n)^2] exists for any (and does not depend on ), and if , then converges in distribution to the normal distribution . In this work we prove that, for any initial distribution satisfying , - If is dominated by with , then the rate of convergence in the c.l.t is . - If is dominated by with , then satisfies a local limit theorem under a usual non-arithmeticity assumption
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