English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the zaia usage as premium feature

ZAIA

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the access of premium content as premium feature

Premium Content

Zendy Tools

Zendy Open

0. Introduction The structure and solidarity properties of general Markov chains satisfying the measure-theoretic condition of 59-irreducibility, for some <p, are now well known (see, for example, Orey [8], Revuz [11]), and show a striking similarity to these properties for irreducible Markov chains on a countable state space. If the state space is topological then the condition of <pirreducibility is somewhat unnatural; however, it has so far proved difficult to achieve results of the same strength when 53-irreducibility is replaced by other, more topologically motivated, conditions. Several authors have given results bearing some resemblance to the '99-solidarity' properties for Markov chains whose transition probabilities are weakly continuous (see, for example, Foguel [2], Jamison [5], Rosenblatt [13, 14]). In this paper we propose a different form of continuity condition, namely the existence of a continuous component in the transition mechanism of the chain. Our condition is automatically satisfied by strongly Feller chains, i.e. in the special case where the transition probabilities are strongly continuous; but in general it is much weaker than the strong Feller condition. In § 4 we shall show that if this 'continuous component' is non-trivial in suitable ways then certain local topological irreducibility, recurrence or positivity conditions imply the corresponding ^-solidarity properties. These results are analogous to those known for 'spread-out' random walks, and we show in §5 that in fact our continuous component exists for random walks if and only if the increment distribution is spread-out; hence our condition can be seen as an exact generalization of this random walk concept for more general Markov chains. In § 6 we turn to chains which do not necessarily have any topological irreducibility structure. The main theorem of this section, and of the whole paper, states that, when the chain has a suitably non-trivial continuous component and the topology is Tx with a countable basis, the state space admits a decomposition into a countable number of stochastically closed Harris sets (i.e. the restriction of the chain to each

Address

About

Learn

Download the Zendy App

Discover

Explore