Let {X_n, n = 0, 1, 2, ...} denote a Markov chain on a general state space and let f be a nonnegative function. The purpose of this paper is to present conditions which will imply that f(X_n) tends to 0 a.s., as n tends to infinity. As an application we obtain a result on synchronisation for random dynamical systems. At the end of the paper, we also present a result on convergence in distribution for Markov chains on general state spaces, thereby generalising a similar result for Markov chains on compact metric spaces.

AMS 2010 Subject Classification: 60J05, 60F15, 60F05.

A very simple specific case of a Polya urn scheme is as follows. At each trial one draws a ball from an urn with balls of two different colours. Then, one looks at the ball, and returns the ball to the urn together with another ball of the same colour. Then one makes another draw. Et cetera. At the first draw there is one ball of each colour.

The rechargeable Polya urn scheme is essentially the same except that between each draw there is a fixed probability that the process starts over with two balls in the urn having different colours.

Now, for n = 1,2, ..., let B(n) and G(n) denote respectively the number of blue and yellow balls in the urn and let Y(n) denote the colour of the ball drawn at the nth draw. Further let Z(n) denote the probability distribution of (B(n),G(n)) given that we have observed Y(m), from m = 1 to m = n. In this note we prove that the sequence Z(1),Z(2),.... converges in distribution.

Consider a Hidden Markov Model (HMM) such that both the state space and the observation space are complete, separable, metric spaces and for which both the transition probability function (tr.pr.f.) determining the hidden Markov chain of the HMM and the tr.pr.f. determining the observation sequence of the HMM have densities. Such HMMs are called fully dominated. In this paper we consider a subclass of fully dominated HMMs which we call regular. A fully dominated, regular HMM induces a tr.pr.f. on the set of probability density functions on the state space which we call the filter kernel induced by the HMM and which can be interpreted as the Markov kernel associated to the sequence of conditional state distributions. We show that if the underlying hidden Markov chain of the fully dominated, regular HMM is strongly ergodic and a certain coupling condition is fulfilled, then, in the limit, the distribution of the conditional distribution becomes independent of the initial distribution of the hidden Markov chain and, if also the hidden Markov chain is uniformly ergodic, then the distributions tend towards a limit distribution. In the last part of the paper, we present some more explicit conditions, implying that the coupling condition mentioned above is satisfied.

Let g(x) = x/2 + 17/30 (mod 1), let 𝝃_i i = 1,2, … to be a sequence of independent, identically distributed random variables with uniform distribution on the interval [0,1/15], define g_i(x) = g(x) + 𝝃_i (mod 1) and for n = 1,2, …, define gⁿ (x) = g_n(g_n-1(…(g₁(x))…)). For x ϵ [0,1) let μ_n,x denote the distribution of gⁿ(x). The purpose of this note is to show that there exists a unique probability measure μ, such that, for all x ϵ [0,1); μ_n,x tends to μ as n → ∞. This contradicts a claim by Lasota and Mackey from 1987 stating that the process has an asymptotic three-periodicity.

Let {X(n), n=0,1,2,...} denote a Markov chain on a general state space and let f be a nonnegative function. The purpose of this paper is to present conditions which will imply that f(X(n)) tends to 0 a.s., as n tends to infinity. As an application we obtain a result on "synchronisation for random dynamical systems". At the end of the paper we also present a result on  "convergence in distribution" for random dynamical system on complete, separable, metric spaces, a result, which is a generalisation of  a similar result for random dynamical systems on compact, metric spaces.

A Hidden Markov Model generates two basic stochastic processes, a Markov chain, which is hidden, and an observation sequence. The filtering process of a Hidden Markov Model is, roughly speaking, the sequence of conditional distributions of the hidden Markov chain that is obtained as new observations are received.

It is well-known, that the filtering process itself, is also a Markov chain. A classical, theoretical problem is to find conditions which implies that the distributions of the filtering process converge towards a unique limit measure.

This problem goes back to a paper of D Blackwell for the case when the Markov chain takes its values in a finite set and it goes back to a paper of H Kunita for the case when the state space of the Markov chain is a compact Hausdor space.

Recently, due to work by F Kochmann, J Reeds, P Chigansky and R van Handel, a necessary and sucient condition for the convergence of the distributions of the filtering process has been found for the case when the state space is finite. This condition has since been generalised to the case when the state space is denumerable.

In this paper we generalise some of the previous results on convergence in distribution to the case when the Markov chain and the observation sequence of a Hidden Markov Model take their values in complete, separable, metric spaces; it has though been necessary to assume that both the transition probability function of the Markov chain and the transition probability function that generates the observation sequence have densities.

Let S be a denumerable state space and let P be a transition probability matrix on S. If a denumerable set M of nonnegative matrices is such that the sum of the matrices is equal to P, then we call M a partition of P. Let K denote the set of probability vectors on S. With every partition M of P we can associate a transition probability function PM on K defined in such a way that if p is an element of K and M is an element of M are such that parallel to pM parallel to andgt; 0, then, with probability parallel to pM parallel to, the vector p is transferred to the vector PM/parallel to pM parallel to. Here parallel to . parallel to denotes the l(1)-norm. In this paper we investigate the convergence in distribution for Markov chains generated by transition probability functions induced by partitions of transition probability matrices. The main motivation for this investigation is the application of the convergence results obtained to filtering processes of partially observed Markov chains with denumerable state space.

Let S be a denumerable state space and let P be a transition probability matrix on S. If a denumerable set M of nonnegative matrices is such that the sum of the matrices is equal to P, then we call M a partition of P.

Let K denote the set of probability vectors on S. With every partition M of P we can associate a transition probability function P _M on K defined in such a way that if p ∈ K and M ∈ M are such that ‖pM‖ > 0, then, with probability ‖pM‖, the vector p is transferred to the vector pM/‖pM‖. Here ‖·‖ denotes the l ₁-norm.

In this paper we investigate the convergence in distribution for Markov chains generated by transition probability functions induced by partitions of transition probability matrices. The main motivation for this investigation is the application of the convergence results obtained to filtering processes of partially observed Markov chains with denumerable state space.