498,090 research outputs found
Degeneracy Implies Non-abelian Statistics
A non-abelian anyon can only occur in the presence of ground state degeneracy
in the plane. It is conceivable that for some strange anyon with quantum
dimension that the resulting representations of all -strand braid
groups are overall phases, even though the ground state manifolds for
such anyons in the plane are in general Hilbert spaces of dimensions . We
observe that degeneracy is all that is needed: for an anyon with quantum
dimension the non-abelian statistics cannot all be overall phases on the
degeneracy ground state manifold. Therefore, degeneracy implies non-abelian
statistics, which justifies defining a non-abelian anyon as one with quantum
dimension . Since non-abelian statistics presumes degeneracy, degeneracy is
more fundamental than non-abelian statistics.Comment: State the main result as a theorem and add several clarification
Merging for inhomogeneous finite Markov chains, part II: Nash and log-Sobolev inequalities
We study time-inhomogeneous Markov chains with finite state spaces using Nash
and logarithmic-Sobolev inequalities, and the notion of -stability. We
develop the basic theory of such functional inequalities in the
time-inhomogeneous context and provide illustrating examples.Comment: Published in at http://dx.doi.org/10.1214/10-AOP572 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Frequentist optimality of Bayesian wavelet shrinkage rules for Gaussian and non-Gaussian noise
The present paper investigates theoretical performance of various Bayesian
wavelet shrinkage rules in a nonparametric regression model with i.i.d. errors
which are not necessarily normally distributed. The main purpose is comparison
of various Bayesian models in terms of their frequentist asymptotic optimality
in Sobolev and Besov spaces. We establish a relationship between
hyperparameters, verify that the majority of Bayesian models studied so far
achieve theoretical optimality, state which Bayesian models cannot achieve
optimal convergence rate and explain why it happens.Comment: Published at http://dx.doi.org/10.1214/009053606000000128 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions
Most previous contributions to BSDEs, and the related theories of nonlinear
expectation and dynamic risk measures, have been in the framework of continuous
time diffusions or jump diffusions. Using solutions of BSDEs on spaces related
to finite state, continuous time Markov chains, we develop a theory of
nonlinear expectations in the spirit of [Dynamically consistent nonlinear
evaluations and expectations (2005) Shandong Univ.]. We prove basic properties
of these expectations and show their applications to dynamic risk measures on
such spaces. In particular, we prove comparison theorems for scalar and vector
valued solutions to BSDEs, and discuss arbitrage and risk measures in the
scalar case.Comment: Published in at http://dx.doi.org/10.1214/09-AAP619 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Characterization of the optimal risk-Sensitive average cost in finite controlled Markov chains
This work concerns controlled Markov chains with finite state and action
spaces. The transition law satisfies the simultaneous Doeblin condition, and
the performance of a control policy is measured by the (long-run)
risk-sensitive average cost criterion associated to a positive, but otherwise
arbitrary, risk sensitivity coefficient. Within this context, the optimal
risk-sensitive average cost is characterized via a minimization problem in a
finite-dimensional Euclidean space.Comment: Published at http://dx.doi.org/10.1214/105051604000000585 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
A method for inferring hierarchical dynamics in stochastic processes
Complex systems may often be characterized by their hierarchical dynamics. In
this paper do we present a method and an operational algorithm that
automatically infer this property in a broad range of systems; discrete
stochastic processes. The main idea is to systematically explore the set of
projections from the state space of a process to smaller state spaces, and to
determine which of the projections that impose Markovian dynamics on the
coarser level. These projections, which we call Markov projections, then
constitute the hierarchical dynamics of the system. The algorithm operates on
time series or other statistics, so a priori knowledge of the intrinsic
workings of a system is not required in order to determine its hierarchical
dynamics. We illustrate the method by applying it to two simple processes; a
finite state automaton and an iterated map.Comment: 16 pages, 12 figure
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