2,252 research outputs found
A classification of invariant distributions and convergence of imprecise Markov chains
We analyse the structure of imprecise Markov chains and study their
convergence by means of accessibility relations. We first identify the sets of
states, so-called minimal permanent classes, that are the minimal sets capable
of containing and preserving the whole probability mass of the chain. These
classes generalise the essential classes known from the classical theory. We
then define a class of extremal imprecise invariant distributions and show that
they are uniquely determined by the values of the upper probability on minimal
permanent classes. Moreover, we give conditions for unique convergence to these
extremal invariant distributions
Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree
We continue our study of the full set of translation-invariant splitting
Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for
the -state Potts model on a Cayley tree. In our previous work \cite{KRK} we
gave a full description of the TISGMs, and showed in particular that at
sufficiently low temperatures their number is .
In this paper we find some regions for the temperature parameter ensuring
that a given TISGM is (non-)extreme in the set of all Gibbs measures.
In particular we show the existence of a temperature interval for which there
are at least extremal TISGMs.
For the Cayley tree of order two we give explicit formulae and some numerical
values.Comment: 44 pages. To appear in Random Structures and Algorithm
The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains
We introduce the cluster index of a multivariate regularly varying stationary
sequence and characterize the index in terms of the spectral tail process. This
index plays a major role in limit theory for partial sums of regularly varying
sequences. We illustrate the use of the cluster index by characterizing
infinite variance stable limit distributions and precise large deviation
results for sums of multivariate functions acting on a stationary Markov chain
under a drift condition
Concentration inequalities for dependent Random variables via the martingale method
The martingale method is used to establish concentration inequalities for a
class of dependent random sequences on a countable state space, with the
constants in the inequalities expressed in terms of certain mixing
coefficients. Along the way, bounds are obtained on martingale differences
associated with the random sequences, which may be of independent interest. As
applications of the main result, concentration inequalities are also derived
for inhomogeneous Markov chains and hidden Markov chains, and an extremal
property associated with their martingale difference bounds is established.
This work complements and generalizes certain concentration inequalities
obtained by Marton and Samson, while also providing different proofs of some
known results.Comment: Published in at http://dx.doi.org/10.1214/07-AOP384 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Entropic repulsion and lack of the -measure property for Dyson models
We consider Dyson models, Ising models with slow polynomial decay, at low
temperature and show that its Gibbs measures deep in the phase transition
region are not -measures. The main ingredient in the proof is the occurrence
of an entropic repulsion effect, which follows from the mesoscopic stability of
a (single-point) interface for these long-range models in the phase transition
region.Comment: 22 pages, 4 figure
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