@inbook{1e4b736923e64c298e3ab8d981a3ca82,
title = "Small Sets, Irreducibility, and Aperiodicity",
abstract = "So far, we have considered only atomic and discrete Markov chains. When the state space is not discrete, many Markov chains do not admit accessible atoms. Recall that a set C is an atom if each time the chain visits C, it regenerates, i.e., it leaves C under a probability distribution that is constant over C. If the state space does not possess an atom, we may require instead that the chain restart anew from C with some fixed probability (strictly less than one) that is constant over C. Then this property is satisfied by many more Markov chains. Such sets will be called small sets. The purpose of this chapter is to provide the first basic properties of Markov kernels that admit accessible small sets.",
author = "Randal Douc and Eric Moulines and Pierre Priouret and Philippe Soulier",
note = "Publisher Copyright: {\textcopyright} 2018, Springer Nature Switzerland AG.",
year = "2018",
month = jan,
day = "1",
doi = "10.1007/978-3-319-97704-1\_9",
language = "English",
series = "Springer Series in Operations Research and Financial Engineering",
publisher = "Springer Nature",
pages = "191--220",
booktitle = "Springer Series in Operations Research and Financial Engineering",
}