# “Small sets” in Markov chains

I came across a definition for a "small set" (of the state space) $A \subset \Omega$: there exists a $\delta > 0$ and a measure $\mu$ such that $p^{(k)}(x, \cdot) \geq \delta \mu (\cdot)$ for every $x \in A$. In this case, they say that $A$ has lag $k$.

I have no intuition for this and I can't find anything anywhere that explains this with some examples. Can anyone tell me what it means? Why is it important?

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This definition might depend on the specific context. Can you give the reference of the paper or textbook you are reading? – Srivatsan Nov 26 '11 at 18:48
Since the word "small" is too generic, my guess is that that the definition is not meant to make sense in isolation. Perhaps the author wanted to define a term to be used only in that particular book/chapter/paper. – Srivatsan Nov 26 '11 at 18:55
@Srivatsan as Didier wrote, such concept is indeed often used in the theory of Markov Chains, as well as petite sets which happen to be different. – Ilya Nov 26 '11 at 19:14

This is actually an important and much used concept in the study of Markov chains. The number $\delta$ is meaningful because one assumes $\mu$ is a probability measure (and not only a measure). Then $\delta$ is used to evaluate the rate of the loss of memory of the initial state by the chain.

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I wonder if you can advise a book where such concepts as small sets, petite sets etc. are clearly explained, not only defined. – Ilya Nov 26 '11 at 19:11
A book often referred to in this context is the monography General Irreducible Markov Chains and Non-Negative Operators by Elsa Nummelin (unfortunately, I cannot be more precise, having not read it myself). – Did Nov 26 '11 at 19:29
Thank you, I will check it out in my library. – Ilya Nov 26 '11 at 19:37
Actually I just skimmed through a student's research project, written under the supervision of Jeffrey Rosenthal, which explains this stuff quite competently. Search for Olga Chilina, Spring 2006 on Rosenthal's website. – Did Nov 26 '11 at 19:39
Thanks for the link. Have you read by any chance new edition of 'Markov Chains and Stochastic Stability' - if it's worth to read after I read the first edition? There is no new version in my library, so I was thinking to buy it (and that's not the only one book I would like to buy) – Ilya Nov 26 '11 at 19:49

Yes you can find a lot related topic at Jeffery Rosenthal's paper, especially the "General State Markov chain and MCMC algorithm".

Another useful reference is Nummelin's paper called "MC for MCMC". This is more intuitively than Rosenthal and when I write paper about this topic this summer, I find it is really useful.

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please refer to this paper, it discuss a lot for small set:google.com/… – abrocod Aug 5 '12 at 20:48