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5 Bulletin of the Polish Academy of Sciences. Mathematics

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An application of skew product maps to Markov chains

Kowalski Z. S.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 1

35-41

By using the skew product definition of a Markov chain we obtain the following results: (a) Every k-step Markov chain is a quasi-Markovian process. (b) Every piecewise linear map with a Markovian partition defines a Markov chain forevery absolutely continuous invariant measure. (c) Satisfying the Chapman-Kolmogorov equation is not sufficient for a process to be quasi-Markovian.

Mixing but not exact quasi-Markovian process

Sachse P.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 4

365-379

Mixing Smorodinsky-Adams map [2] with q_n = n^2, is shown to satisfy, with respect to the time-zero partition, the d+ > 1/16 property. Basing on a general theorem from [7], an explicite example of a mixing but not exact quasi-Markovian process is given. The d+ > O property is studied: although defined for a process, it is shown to be a property of the transformation itself. All mixing rank one transformations are shown to possess this property, which strenghtens the result from [3]. Several other transformations are observed not to satisfy this property (von Neumann-Kakutani's map, irrational rotations, Feldman's map). A rank one weakly mixing example with d+ = O is also given.

Lightly but not partially mixing quasi-Markovian process

Sachse P.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 4

381-391

A lightly and not partially mixing rank one transformation, considered in [2], is shown to satisfy the d+ > O property. This permits to construct an explicite example of a lightly and not partially mixing quasi-Markovian process, as well as a piecewise monotone and continuous, expanding interval map, lightly and not partially mixing, which complements the results of [1] and [6].

Very weak Bernoulli processes which are not quasi-Markovian ones

Kowalski Z. S.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 1

89-92

We present an example of the map T with Markovian generator Q such that the process (T, Q) is not quasi-Markovian but the natural extension of T to the automorphism is Bernoulli one.

Quasi-eigenfunctions and quasi-Markovian processes

Kowalski Z., Sachse P.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 2

133-142

We try to characterize q.m.p. using quasi-eigenfunctions and generating Markov partitions. As a result we get some examples of processes which are not q.m.p. and an example of a process satisfying some weaker conditions, than the quasi-Markovian one.