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Convergence diagnosis to stationary distribution in MCMC methods via atoms and renewal sets

Romaniuk M.

Control and Cybernetics

2008

Vol. 37, no 1

205-229

MCMC setups are one of the best known methods for conducting computer simulations useful in such areas as statistics, physics, biology, etc. However, to obtain appropriate solutions, the additional convergence diagnosis must be applied for Markov Chain trajectory generated by the algorithm. We present the method for dealing with this problem based on features of so called "secondary" chain (the chain with specially selected state space). The secondary chain is created from the initial chain by picking only some observations connected with atoms or renewal sets. In this paper we focus on finding the moment when the simulated chain is close enough to the stationary distribution of the Markov chain. The discussed method has some appealing properties, like high degree of diagnosis automation. Apart from theoretical lemmas and a more heuristic approach, the examples of application are also provided.

On some method for diagnosing convergence in MCMC setups via atoms and renewal sets

Romaniuk M.

Control and Cybernetics

2007

Vol. 36, no 4

985-1008

MCMC setups are among the best known methods for conducting computer simulations necessary in statistics, physics, biology, etc. However, to obtain appropriate solutions, additional convergence diagnosis must be applied for trajectory generated by Markov Chain. In the paper we present, the method for dealing with this problem, based on features of so called "secondary" chain (the chain with specially selected state space). The secondary chain is created from the initial chain by picking only some observations connected with atoms or renewal sets. The discussed method has some appealing properties, like high degree of diagnosis automation. Apart from theoretical lemmas, the example of application is also provided.

On the fractional record values

Bieniek M., Szynal D.

Probability and Mathematical Statistics

2004

Vol. 24, Fasc. 1

27--46

We define the record-values process which may be considered as the collection of record values with non-integer or fractional indices. The alternative con-struction from the sample as well as the basic properties of the defined process are shown.

An abstract Lagrangian process related to convection-diffusion of a passive tracer in a Markovian flow

Komorowski T.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 4

413-427

In this paper we consider a motion of a tracer particle in a time Markovian random environment. We show that an abstract valued process describing the environment from the point of view of the moving particle, the so-called Lagrangian dynamics, is also Markovian. We show several properties of the L[sup 2]-semigroup corresponding to this process.