16,409 research outputs found
A constructive version of Birkhoff's ergodic theorem for Martin-L\"of random points
A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite
binary sequence {\omega} and an effectively open set A of measure less than 1,
some tail of {\omega} is not in A. We first prove several results in the same
spirit and generalize them via an effective version of a weak form of
Birkhoff's ergodic theorem. We then use this result to get a stronger form of
it, namely a very general effective version of Birkhoff's ergodic theorem,
which improves all the results previously obtained in this direction, in
particular those of V'Yugin, Nandakumar and Hoyrup, Rojas.Comment: Improved version of the CiE'10 paper, with the strong form of
Birkhoff's ergodic theorem for random point
Control Variates for Reversible MCMC Samplers
A general methodology is introduced for the construction and effective
application of control variates to estimation problems involving data from
reversible MCMC samplers. We propose the use of a specific class of functions
as control variates, and we introduce a new, consistent estimator for the
values of the coefficients of the optimal linear combination of these
functions. The form and proposed construction of the control variates is
derived from our solution of the Poisson equation associated with a specific
MCMC scenario. The new estimator, which can be applied to the same MCMC sample,
is derived from a novel, finite-dimensional, explicit representation for the
optimal coefficients. The resulting variance-reduction methodology is primarily
applicable when the simulated data are generated by a conjugate random-scan
Gibbs sampler. MCMC examples of Bayesian inference problems demonstrate that
the corresponding reduction in the estimation variance is significant, and that
in some cases it can be quite dramatic. Extensions of this methodology in
several directions are given, including certain families of Metropolis-Hastings
samplers and hybrid Metropolis-within-Gibbs algorithms. Corresponding
simulation examples are presented illustrating the utility of the proposed
methods. All methodological and asymptotic arguments are rigorously justified
under easily verifiable and essentially minimal conditions.Comment: 44 pages; 6 figures; 5 table
Ultraproducts and metastability
Given a convergence theorem in analysis, under very general conditions a
model-theoretic compactness argument implies that there is a uniform bound on
the rate of metastability. We illustrate with three examples from ergodic
theory
Upcrossing inequalities for stationary sequences and applications
For arrays of random variables that are
stationary in an appropriate sense, we show that the fluctuations of the
process can be bounded in terms of a measure of the
``mean subadditivity'' of the process . We derive
universal upcrossing inequalities with exponential decay for Kingman's
subadditive ergodic theorem, the Shannon--MacMillan--Breiman theorem and for
the convergence of the Kolmogorov complexity of a stationary sample.Comment: Published in at http://dx.doi.org/10.1214/09-AOP460 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Stochastic homogenization of interfaces moving with changing sign velocity
We are interested in the averaged behavior of interfaces moving in stationary
ergodic environments, with oscillatory normal velocity which changes sign. This
problem can be reformulated, using level sets, as the homogenization of a
Hamilton-Jacobi equation with a positively homogeneous non-coercive
Hamiltonian. The periodic setting was earlier studied by Cardaliaguet, Lions
and Souganidis (2009). Here we concentrate in the random media and show that
the solutions of the oscillatory Hamilton-Jacobi equation converge in
-weak to a linear combination of the initial datum and the
solutions of several initial value problems with deterministic effective
Hamiltonian(s), determined by the properties of the random media
Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear
operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n
x. We prove the following variational inequality in the case where T is power
bounded from above and below: for any increasing sequence (t_k)_{k in N} of
natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p,
where the constant C depends only on p and the modulus of uniform convexity.
For T a nonexpansive operator, we obtain a weaker bound on the number of
epsilon-fluctuations in the sequence. We clarify the relationship between
bounds on the number of epsilon-fluctuations in a sequence and bounds on the
rate of metastability, and provide lower bounds on the rate of metastability
that show that our main result is sharp
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