3,456 research outputs found
Imprecise continuous-time Markov chains : efficient computational methods with guaranteed error bounds
Imprecise continuous-time Markov chains are a robust type of continuous-time Markov chains that allow for partially specified time-dependent parameters. Computing inferences for them requires the solution of a non-linear differential equation. As there is no general analytical expression for this solution, efficient numerical approximation methods are essential to the applicability of this model. We here improve the uniform approximation method of Krak et al. (2016) in two ways and propose a novel and more efficient adaptive approximation method. For ergodic chains, we also provide a method that allows us to approximate stationary distributions up to any desired maximal error
Geometric ergodicity of the Random Walk Metropolis with position-dependent proposal covariance
We consider a Metropolis-Hastings method with proposal kernel
, where is the current state. After discussing
specific cases from the literature, we analyse the ergodicity properties of the
resulting Markov chains. In one dimension we find that suitable choice of
can change the ergodicity properties compared to the Random Walk
Metropolis case , either for the better or worse. In
higher dimensions we use a specific example to show that judicious choice of
can produce a chain which will converge at a geometric rate to its
limiting distribution when probability concentrates on an ever narrower ridge
as grows, something which is not true for the Random Walk Metropolis.Comment: 15 pages + appendices, 4 figure
On the long time behavior of the TCP window size process
The TCP window size process appears in the modeling of the famous
Transmission Control Protocol used for data transmission over the Internet.
This continuous time Markov process takes its values in , is
ergodic and irreversible. It belongs to the Additive Increase Multiplicative
Decrease class of processes. The sample paths are piecewise linear
deterministic and the whole randomness of the dynamics comes from the jump
mechanism. Several aspects of this process have already been investigated in
the literature. In the present paper, we mainly get quantitative estimates for
the convergence to equilibrium, in terms of the Wasserstein coupling
distance, for the process and also for its embedded chain.Comment: Correction
Large deviation asymptotics and control variates for simulating large functions
Consider the normalized partial sums of a real-valued function of a
Markov chain, The
chain takes values in a general state space ,
with transition kernel , and it is assumed that the Lyapunov drift condition
holds: where , , the set is small and dominates . Under these
assumptions, the following conclusions are obtained: 1. It is known that this
drift condition is equivalent to the existence of a unique invariant
distribution satisfying , and the law of large numbers
holds for any function dominated by :
2. The lower error
probability defined by , for , ,
satisfies a large deviation limit theorem when the function satisfies a
monotonicity condition. Under additional minor conditions an exact large
deviations expansion is obtained. 3. If is near-monotone, then
control-variates are constructed based on the Lyapunov function , providing
a pair of estimators that together satisfy nontrivial large asymptotics for the
lower and upper error probabilities. In an application to simulation of queues
it is shown that exact large deviation asymptotics are possible even when the
estimator does not satisfy a central limit theorem.Comment: Published at http://dx.doi.org/10.1214/105051605000000737 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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