-IrreducibilityLet L(x, A) denote the probability that a Markov chain started at x ever hits the set A. A rather complicated formula for L(x, A) is given by Meyn and Tweedie (1993, p. 72) but is not of interest here.
A Markov chain on a measurable space 
is
-irreducible if is a nonzero measure
on such that L(x, B) ;SPMgt; 0
for all
and all 
such that 
.
If a chain is -irreducible for any
then it is also 
-irreducible if is a stationary
distribution for the chain (Meyn and Tweedie, 1993, Proposition 4.4.2 and
Theorem 10.4.9), moreover must then be the unique stationary
distribution. The point of the weaker notion of -irreducibility
is that it allows one to check fewer sets.
Let 
denote the probability 
that the chain started at x is in B after n steps. Then an equivalent
definition of -irreducibility is that for every and
every such that there exists an n
such that 
.