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Winter 2010
Pstat160a
Handout MC#3
Applications of making a state absorbing
1. General Remarks
If a Markov Chain
Y
n
has an absorbing state
a
and transition matrix
P
y
whose
n
th
power is
P
n
y
, then
P
n
y
(
i,a
) represent
not only the usual
n
steps probability, but also the probability that
Y
n
reached
a
in
n
or less steps
, since the chain
“get stuck” in
a
. Similarly,
P
n
y
(
i,j
)
, i,j
6
=
a
represent the probability that
Y
n
never visited
a
in the ﬁrst
N
steps. Now
suppose that the chain
X
n
has the same transition probability than
Y
n
, except that
a
is not absorbing. The chains
X
n
and
Y
n
behave the same way until they reach the state
a
, that is their transition probabilities are identical up to that
point.
2. Visit to a state in less than
n
steps
Starting with a Markov chain with matrix
P
, suppose we want to ﬁnd the probability that the chain visit a particular
state in
n
steps or less. It should be clear by taking complement that this will give us also the probability that the chain
never visit that state in the ﬁrst
n
steps.
The basic idea is to make this state absorbing (call it
a
) and leave the rest of the chain intact. The resulting new chain
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