303 research outputs found
Exponential chi squared distributions in infinite ergodic theory
We prove distributional limit theorems for random walk adic transformations
obtaining ergodic distributional limits of exponential chi squared form.Comment: Keywords: Infinite ergodic theory, distributional convergence, random
walk adic transformatio
Heavy tailed solutions of multivariate smoothing transforms
Let be a fixed integer and a random element of
. We consider solutions of multivariate smoothing
transforms, i.e. random variables satisfying R \eqdist \sum_{i=1}^N C_i
R_i +Q where \eqdist denotes equality in distribution, and are independent identically distributed -valued random variables,
and independent of . We briefly review conditions for the
existence of solutions, and then study their asymptotic behaviour. We show that
under natural conditions, these solutions exhibit heavy tails. Our results also
cover the case of complex valued weights .Comment: 35 page
Symmetric spaces of higher rank do not admit differentiable compactifications
Any nonpositively curved symmetric space admits a topological
compactification, namely the Hadamard compactification. For rank one spaces,
this topological compactification can be endowed with a differentiable
structure such that the action of the isometry group is differentiable.
Moreover, the restriction of the action on the boundary leads to a flat model
for some geometry (conformal, CR or quaternionic CR depending of the space).
One can ask whether such a differentiable compactification exists for higher
rank spaces, hopefully leading to some knew geometry to explore. In this paper
we answer negatively.Comment: 13 pages, to appear in Mathematische Annale
Flag Structures on Seifert Manifolds
We consider faithful projective actions of a cocompact lattice of SL(2,R) on
the projective plane, with the following property: there is a common fixed
point, which is a saddle fixed point for every element of infinite order of the
the group. Typical examples of such an action are linear actions, ie, when the
action arises from a morphism of the group into GL(2,R), viewed as the group of
linear transformations of a copy of the affine plane in RP^{2}. We prove that
in the general situation, such an action is always topologically linearisable,
and that the linearisation is Lipschitz if and only if it is projective. This
result is obtained through the study of a certain family of flag structures on
Seifert manifolds. As a corollary, we deduce some dynamical properties of the
transversely affine flows obtained by deformations of horocyclic flows. In
particular, these flows are not minimal.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol5/paper7.abs.htm
On the embeddability of certain infinitely divisible probability measures on Lie groups
We describe certain sufficient conditions for an infinitely divisible
probability measure on a class of connected Lie groups to be embeddable in a
continuous one-parameter convolution semigroup of probability measures.
(Theorem 1.3). This enables us in particular to conclude the embeddability of
all infinitely divisible probability measures on certain Lie groups, including
the so called Walnut group (Corollary 1.5). The embeddability is concluded also
under certain other conditions (Corollary 1.4 and Theorem 1.6).Comment: 24 page
Contracting automorphisms and L^p-cohomology in degree one
We characterize those Lie groups, and algebraic groups over a local field of
characteristic zero, whose first reduced L^p-cohomology is zero for all p>1,
extending a result of Pansu. As an application, we obtain a description of
Gromov-hyperbolic groups among those groups. In particular we prove that any
non-elementary Gromov-hyperbolic algebraic group over a non-Archimedean local
field of zero characteristic is quasi-isometric to a 3-regular tree. We also
extend the study to semidirect products of a general locally compact group by a
cyclic group acting by contracting automorphisms.Comment: 27 pages, no figur
Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment
The objective of the present paper is to establish exponential large
deviation inequalities, and to use them to show exponential concentration
inequalities for the free energy of a polymer in general random environment,
its rate of convergence, and an expression of its limit value in terms of those
of some multiplicative cascades.Comment: 25 page
A local limit theorem for a transient chaotic walk in a frozen environment
This paper studies particle propagation in a one-dimensional inhomogeneous
medium where the laws of motion are generated by chaotic and deterministic
local maps. Assuming that the particle's initial location is random and
uniformly distributed, this dynamical system can be reduced to a random walk in
a one-dimensional inhomogeneous environment with a forbidden direction. Our
main result is a local limit theorem which explains in detail why, in the long
run, the random walk's probability mass function does not converge to a
Gaussian density, although the corresponding limiting distribution over a
coarser diffusive space scale is Gaussian.Comment: 26 pages, 2 figures. To appear in Stochastic Processes and their
Application
Extreme values and fat tails of multifractal fluctuations
In this paper we discuss the problem of the estimation of extreme event
occurrence probability for data drawn from some multifractal process. We also
study the heavy (power-law) tail behavior of probability density function
associated with such data. We show that because of strong correlations,
standard extreme value approach is not valid and classical tail exponent
estimators should be interpreted cautiously. Extreme statistics associated with
multifractal random processes turn out to be characterized by non
self-averaging properties. Our considerations rely upon some analogy between
random multiplicative cascades and the physics of disordered systems and also
on recent mathematical results about the so-called multifractal formalism.
Applied to financial time series, our findings allow us to propose an unified
framemork that accounts for the observed multiscaling properties of return
fluctuations, the volatility clustering phenomenon and the observed ``inverse
cubic law'' of the return pdf tails
Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices
Given a sequence of i.i.d.\ random variables with
generic copy , we consider the random
difference equation (RDE) , and assume
the existence of such that \lim_{n \to \infty}(\E{\norm{M_1 ...
M_n}^\kappa})^{\frac{1}{n}} = 1 . We prove, under suitable assumptions, that
the sequence , appropriately normalized, converges in
law to a multidimensional stable distribution with index . As a
by-product, we show that the unique stationary solution of the RDE is
regularly varying with index , and give a precise description of its
tail measure. This extends the prior work http://arxiv.org/abs/1009.1728v3 .Comment: 15 page
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