Homework Help: Show all other states are transient in Markov chain

1. Jun 26, 2011

xentity1x

Let X be a Markov chain with a state s that is absorbing, i.e. pss(1) = 1. All other states
communicate with s i.e. i → s for all states i ∈ S. Show that all states in S except s are
transient.

I understand this intuitively, but I'm not really sure how to start the proof.

2. Jun 26, 2011

lanedance

write down a transition matrix (which can be reasonably arbitrary, make the absorbing state the first for simplicity) and consider the applying it again and again, and think what happens in the limit...

3. Jun 27, 2011

xentity1x

Ok so I wrote down the transition matrix with s=1.
\begin{bmatrix}
\begin{array}{cc}
1 & 0 & ... & 0 \\
p_{21} & p_{22} & ... & p_{2N} \\
\vdots & \vdots & \ddots & \vdots \\
p_{N1} & p_{N2} & ... & p_{NN} \\
\end{array}
\end{bmatrix}

I then raised it to the nth power
\begin{bmatrix}
\begin{array}{cc}
1 & 0 & ... & 0 \\
p_{21}(n) & p_{22}(n) & ... & p_{2N}(n) \\
\vdots & \vdots & \ddots & \vdots \\
p_{N1}(n) & p_{N2}(n) & ... & p_{NN}(n) \\
\end{array}
\end{bmatrix}

So I guess I have to show $$\lim_{n\rightarrow \infty} p_{ij}(n)=0 \hspace{3mm} \forall i,j>1$$

I'm not really to sure where to go from there.

4. Jun 27, 2011

lanedance

not quite $$p_{ij}^{(n)}$$will represent the probability of transitioning from state i at time 0, to state j at time n. So you would expect
$$p_{ij}^{(n)} = 1 / / i=1$$
$$p_{ij}^{(n)} = 0 / / i \neq 1$$

I would start by looking at P^2, P^3 to get a feeling for the form and what happens and try and generalise from there

keep in mind each row of P, P^2 and so forth must sum to one...