# Limit for Markov Chains

1. Mar 11, 2012

### spitz

1. The problem statement, all variables and given/known data

Consider:

$P=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$

Show that $P^n$ has no limit, but that: $A_n=\frac{1}{n+1}(I+P+P^2+\ldots+P^n)$ has a limit.

3. The attempt at a solution

I can see that $P^{EVEN}=\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$ and $P^{ODD}=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$, so a steady state is never reached, but I can't figure out the second part.

Any suggestions?

2. Mar 11, 2012

### Office_Shredder

Staff Emeritus
So for example

I+P+P2+P3+P4+P5 =
[3 3]
[3 3]

and when you divide this by six you get a matrix with all 1/2s. Try adding up some more guys and see what happens