Page 439 - Special Topic Session (STS) - Volume 3

STS555 Patrice Bertail et al.
Roughly speaking, an atom is a set from which all the transition probabilities
are the same. Suppose that X possesses an accessible atom. We define the
sequence of regeneration times (τA(j))j≥1, i.e.

τA = τA(1) = inf{n ≥ 1 : Xn ∈ A}

is the first time when the chain hits the regeneration set A and

τA(j) = inf{n > τA(j − 1), Xn ∈ A} for j ≥ 2

is the j-th visit of the chain to the atom A.
By the strong Markov property, given any initial law ν, the sample paths
can be divided into i.i.d. segments corresponding to the consecutive visits of the
chain to regeneration set A. The blocks of data are of the form:
Bj = (X1+τA(j), · · ·, XτA(j+1)), j ≥ 1

k
and take values in the torus ∪∞k=1E .
In the following, we are interested in steady-state analysis of Markov
chains. More specifically, for a positive recurrent Markov chain if EA(τA) < ∞,
then the unique invariant probability distribution µ is the Pitman’s occupation
measure

We introduce few more pieces of notation: ln = ∑ "{Xi ∈ A} designates
=1
the total number of consecutive visits of the chain to the atom A, thus we
observe ln + 1 data segments. We make the convention that
when τA(ln) = n. We denote by l(Bj) = τA(j + 1) − τA(j), j ≥ 1 the length of
regeneration blocks. From Kac’s theorem it follows that

General Harris Markov chains and the splitting technique
In this framework, we also consider more general classes of Markov chains,
namely positive recurrent Harris Markov chains.
Definition 2. Suppose that X is a ψ-irreducible Markov chain. We say that X is
Harris recurrent iff, starting from any point x ∈ E and any set such that ψ(A) >
0, we have

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