Question Four

A Markov chain with the non$-$negative integers as its state space has transition matrix:

$\left[\begin{array}{cccccccc}0& 1& 0& 0& 0& \text{c}\text{d}\text{o}\text{t}\text{s}& ?& ?\\ \frac{1}{2}& 0& \frac{1}{2}& 0& 0& \text{c}\text{d}\text{o}\text{t}\text{s}& ?& ?\\ 0& \frac{3}{4}& 0& \frac{1}{4}& 0& \text{c}\text{d}\text{o}\text{t}\text{s}& ?& ?\\ 0& 0& \frac{7}{8}& 0& \frac{1}{8}& \text{c}\text{d}\text{o}\text{t}\text{s}& ?& ?\\ 0& 0& 0& \frac{15}{16}& 0& \frac{1}{16}& 0& \text{c}\text{d}\text{o}\text{t}\text{s}\\ ?& \text{c}\text{d}\text{o}\text{t}\text{s}& ?& ?& ?& ?& ?& ?\end{array}\right]$

$($a$)$ Write down the formula for the transition probabilities

$($b$)$ Let the stationary vector be $V_0,V_1,V_2,V_3,$cdots. Express $V_x$ in terms of $V_{x-1}$ and $V_{x+1}$

$($c$)$ Express $V_3$ in terms of $V_0$

$($d$)$ Find $P_{33}^{(2)}$

$($e$)$ Let $G(s)={\displaystyle \underset{j=0}{\overset{}{}}}{V}_j{s}^j.$ Show that $G(s)=p(s)G(\frac{1}{2}s),$ where $p(s)$ is a polynomial in $s.$