937 research outputs found
Mixing and decorrelation in infinite measure: the case of the periodic sinai billiard
We investigate the question of the rate of mixing for observables of a Z
d-extension of a probability preserving dynamical system with good spectral
properties. We state general mixing results, including expansions of every
order. The main part of this article is devoted to the study of mixing rate for
smooth observables of the Z 2-periodic Sinai billiard, with different kinds of
results depending on whether the horizon is finite or infinite. We establish a
first order mixing result when the horizon is infinite. In the finite horizon
case, we establish an asymptotic expansion of every order, enabling the study
of the mixing rate even for observables with null integrals
Mixing rate in infinite measure for Z^d-extension, application to the periodic Sinai billiard
We study the rate of mixing of observables of Z^d-extensions of probability
preserving dynamical systems. We explain how this question is directly linked
to the local limit theorem and establish a rate of mixing for general classes
of observables of the Z^2-periodic Sinai billiard. We compare our approach with
the induction method
Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard
We show how Rio's method [Probab. Theory Related Fields 104 (1996) 255--282]
can be adapted to establish a rate of convergence in in
the multidimensional central limit theorem for some stationary processes in the
sense of the Kantorovich metric. We give two applications of this general
result: in the case of the Knudsen gas and in the case of the Sinai billiard.Comment: Published at http://dx.doi.org/10.1214/105051605000000476 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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)
Back to balls in billiards
We consider a billiard in the plane with periodic configuration of convex
scatterers. This system is recurrent, in the sense that almost every orbit
comes back arbitrarily close to the initial point. In this paper we study the
time needed to get back in an r-ball about the initial point, in the phase
space and also for the position, in the limit when r->0. We establish the
existence of an almost sure convergence rate, and prove a convergence in
distribution for the rescaled return times
Quantitative recurrence in two-dimensional extended processes
Under some mild condition, a random walk in the plane is recurrent. In
particular each trajectory is dense, and a natural question is how much time
one needs to approach a given small neighborhood of the origin. We address this
question in the case of some extended dynamical systems similar to planar
random walks, including \ZZ^2-extension of hyperbolic dynamics. We define a
pointwise recurrence rate and relate it to the dimension of the process, and
establish a convergence in distribution of the rescaled return times near the
origin
Renewal theorems for random walks in random scenery
Random walks in random scenery are processes defined by
, where and
are two independent sequences of i.i.d. random
variables. We suppose that the distributions of and belong to the
normal domain of attraction of strictly stable distributions with index
and respectively. We are interested in the
asymptotic behaviour as goes to infinity of quantities of the form
(when is transient) or
(when is recurrent) where
is some complex-valued function defined on or
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
Persistence exponent for random walk on directed versions of
We study the persistence exponent for random walks in random sceneries (RWRS)
with integer values and for some special random walks in random environment in
including random walks in with random orientations
of the horizontal layers.Comment: 19 page
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