PROBABILITY

Markov chains

 

Invariant probability measures for Markov chains

We consider a (weak) Feller-Markov chain a general Borel space and provide necessary and sufficient conditions for the existence of an invariant probability measure. The problem is viewed as the existence of nonnegative solutions to a linear system in a space of signed mesures. A generalized Farkas theorem of the alternative permits to derive Foster-Lyapunov type (necessary and sufficient) conditions of existence. The issue of uniqueness is also investigated along the same lines.

Invariant probability densities for Markov operators

We consider a Markov operator on a general measure space and provide necessary and sufficient conditions for existence of invariant probability densities. Our conditions are different from (and complement) previous results by Neveu. We also investigate the uniqueness issue. Again, the approach is to consider the existence of nonnegative solutions to a linear system on some L_p space, and to apply a generalized Farkas theorem. A similar approach is applied to the existence of invariant probabilities for Markov processes on a metric space.

The probabilistic multichain Poisson equation

One considers the existence of solutions to the probabilistic multichain Poisson equation, i.e. when the corresponding Markov chain does not have a unique invariant probability distribution. An operator-theoretic approach permits to provide several different characterizations of the solutions.

Ergodic theorems and ergodic decompositions for Markov chains

We consider a Markov chain on a locally compact separable metric space. We characterize the limit in the Birkhoff and Chacon-Ornstein ergodic theorems in terms of the limits of the expected occupations measures. No Feller-like continuity assumption is required and we also provide an ergodic decomposition "a la Yosida".
 

Harris recurrence of Markov chains. Occupation measures.

We have provided a new characterization of positive Harris recurrence for Markov chains on a locally compact separable  metric space. This criterion is stated in terms of the type of convergence of the sequence of expected occupation measures. This criterion also permits to classify the MC's in basically two categories only.

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Measure theory

We have provided a common "uniform principle" behind several types of weak convergences for a sequence of probability measures on a locally compact separable metric space. Included are the weak*, the weak, the setwise convergences as well as the convergence for the weak topologies induced by the Riemann integrable functions and the bounded semi-continous functions. This unifom principle also holds for some weak convergences in Kothe spaces. The Banach lattice property of the underlying spaces is crucial.
We have also provided some results concerning the setwise convergence of a sequence of measures and Fatou and Lebesgue convergence theorems for sequences of measures and mixe sequences of functions and measures.

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