15,078 research outputs found
Conditional ergodicity in infinite dimension
The goal of this paper is to develop a general method to establish
conditional ergodicity of infinite-dimensional Markov chains. Given a Markov
chain in a product space, we aim to understand the ergodic properties of its
conditional distributions given one of the components. Such questions play a
fundamental role in the ergodic theory of nonlinear filters. In the setting of
Harris chains, conditional ergodicity has been established under general
nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional
state spaces are rarely amenable to the classical theory of Harris chains due
to the singularity of their transition probabilities, while topological and
functional methods that have been developed in the ergodic theory of
infinite-dimensional Markov chains are not well suited to the investigation of
conditional distributions. We must therefore develop new measure-theoretic
tools in the ergodic theory of Markov chains that enable the investigation of
conditional ergodicity for infinite dimensional or weak-* ergodic processes. To
this end, we first develop local counterparts of zero-two laws that arise in
the theory of Harris chains. These results give rise to ergodic theorems for
Markov chains that admit asymptotic couplings or that are locally mixing in the
sense of H. F\"{o}llmer, and to a non-Markovian ergodic theorem for stationary
absolutely regular sequences. We proceed to show that local ergodicity is
inherited by conditioning on a nondegenerate observation process. This is used
to prove stability and unique ergodicity of the nonlinear filter. Finally, we
show that our abstract results can be applied to infinite-dimensional Markov
processes that arise in several settings, including dissipative stochastic
partial differential equations, stochastic spin systems and stochastic
differential delay equations.Comment: Published in at http://dx.doi.org/10.1214/13-AOP879 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Perturbation theory for Markov chains via Wasserstein distance
Perturbation theory for Markov chains addresses the question how small
differences in the transitions of Markov chains are reflected in differences
between their distributions. We prove powerful and flexible bounds on the
distance of the th step distributions of two Markov chains when one of them
satisfies a Wasserstein ergodicity condition. Our work is motivated by the
recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the
analysis of big data sets. By using an approach based on Lyapunov functions, we
provide estimates for geometrically ergodic Markov chains under weak
assumptions. In an autoregressive model, our bounds cannot be improved in
general. We illustrate our theory by showing quantitative estimates for
approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings
and stochastic Langevin algorithms.Comment: 31 pages, accepted at Bernoulli Journa
Infinite dimensional entangled Markov chains
We continue the analysis of nontrivial examples of quantum Markov processes.
This is done by applying the construction of entangled Markov chains obtained
from classical Markov chains with infinite state--space. The formula giving the
joint correlations arises from the corresponding classical formula by replacing
the usual matrix multiplication by the Schur multiplication. In this way, we
provide nontrivial examples of entangled Markov chains on , being any infinite dimensional type
factor, a finite interval of , and the bar the von Neumann tensor
product between von Neumann algebras. We then have new nontrivial examples of
quantum random walks which could play a r\^ole in quantum information theory.
In view of applications to quantum statistical mechanics too, we see that the
ergodic type of an entangled Markov chain is completely determined by the
corresponding ergodic type of the underlying classical chain, provided that the
latter admits an invariant probability distribution. This result parallels the
corresponding one relative to the finite dimensional case.
Finally, starting from random walks on discrete ICC groups, we exhibit
examples of quantum Markov processes based on type von Neumann factors.Comment: 16 page
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
Asymptotic Exponential Arbitrage and Utility-based Asymptotic Arbitrage in Markovian Models of Financial Markets
Consider a discrete-time infinite horizon financial market model in which the
logarithm of the stock price is a time discretization of a stochastic
differential equation. Under conditions different from those given in a
previous paper of ours, we prove the existence of investment opportunities
producing an exponentially growing profit with probability tending to
geometrically fast. This is achieved using ergodic results on Markov chains and
tools of large deviations theory.
Furthermore, we discuss asymptotic arbitrage in the expected utility sense
and its relationship to the first part of the paper.Comment: Forthcoming in Acta Applicandae Mathematica
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