1,526 research outputs found
Asymptotic variance of stationary reversible and normal Markov processes
We obtain necessary and sufficient conditions for the regular variation of
the variance of partial sums of functionals of discrete and continuous-time
stationary Markov processes with normal transition operators. We also construct
a class of Metropolis-Hastings algorithms which satisfy a central limit theorem
and invariance principle when the variance is not linear in
Orlicz integrability of additive functionals of Harris ergodic Markov chains
For a Harris ergodic Markov chain , on a general state space,
started from the so called small measure or from the stationary distribution we
provide optimal estimates for Orlicz norms of sums ,
where is the first regeneration time of the chain. The estimates are
expressed in terms of other Orlicz norms of the function (wrt the
stationary distribution) and the regeneration time (wrt the small
measure). We provide applications to tail estimates for additive functionals of
the chain generated by unbounded functions as well as to classical
limit theorems (CLT, LIL, Berry-Esseen)
Nonlinearity and Temporal Dependence
Nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models. We study this link using three measures of temporal dependence: rho-mixing, beta-mixing and alpha-mixing. Stationary diffusions that are rho-mixing have mixing coefficients that decay exponentially to zero. When they fail to be rho-mixing, they are still beta-mixing and alpha-mixing; but coefficient decay is slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state-dependent, Poisson sampling alters the temporal dependence.Diffusion, Strong dependence, Long memory, Poisson sampling, Quadratic forms
Ergodicity of the zigzag process
The zigzag process is a Piecewise Deterministic Markov Process which can be
used in a MCMC framework to sample from a given target distribution. We prove
the convergence of this process to its target under very weak assumptions, and
establish a central limit theorem for empirical averages under stronger
assumptions on the decay of the target measure. We use the classical
"Meyn-Tweedie" approach. The main difficulty turns out to be the proof that the
process can indeed reach all the points in the space, even if we consider the
minimal switching rates
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Large Scale Stochastic Dynamics
In focus are interacting stochastic systems with many components, ranging from stochastic partial differential equations to discrete systems as interacting particles on a lattice moving through random jumps. More specifically one wants to understand the large scale behavior, large in spatial extent but also over long time spans, as entailed by the characterization of stationary measures, effective macroscopic evolution laws, transport of conserved fields, homogenization, self-similar structure and scaling, critical dynamics, dynamical phase transitions, metastability, large deviations, to mention only a few key items
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