In Example 1.4.3 of Norris's book on Markov Chains, the author examines the Gamblers ruin problem using the Strong Markov property and generating functions. Here, the state space is $\{0,1,2,\ldots\}$, and the chain goes from $i\to i+1$ with probability $p$, and $i\to i-1$ with probability $q=1-p$. Let $$ H_j = \inf\{n\geq 0: X_n =j\} $$ denote the first hitting time for value $j$, an define the generating function $$ \phi(s) = \mathbb{E}_1(e^{H_0}) = \sum_{n<\infty} s^n \mathbb{P}_1(H_0=n) $$ for the chain to hit 0, having started at 1.
Norris then writes:
Suppose we start at 2. Apply the strong Markov property at $H_1$ to see that under $\mathbb{P}_2$, conditional on $H_1 < \infty$, we have $H_0 = H_1 + \tilde{H}_0$, where $\tilde{H}_0$, the time taken after $H_1$ to get to 0, is independent of $H_1$ and has the (unconditioned) distribution of $H_1$. So $$ \begin{split} \mathbb{E}_2(s^{H_0}) &=\mathbb{E}_2(s^{H_1}\mid H_1<\infty )\mathbb{E}_2(s^{\tilde H_0}\mid H_1<\infty ) \\ & = \mathbb{E}_2(s^{H_1}1_{H_1<\infty} )\mathbb{E}_2(s^{\tilde H_0}\mid H_1<\infty )\\ & = \mathbb{E}_2(s^{H_1})^2 = \phi(s)^2 \end{split}$$
There are a couple of points here that I'm trying to clarify, and I can't tell if it's me or if Norris is just being a bit sloppy in his presentation.
- In his definition of the generating function, why isn't it defined conditioned on $H_0<\infty$?
- In the first equality, how does Norris introduce the conditioning on $H_1<\infty$?
- How does the author obtain the equality between the second and third line, where one has the indicator function and the other has the conditioning?
All the conditioning is not really necessary if you bear in mind that for $0\le s<1$, $s^{H_1}=0$ on $\{H_1=\infty\}$: $$ \eqalign{ \Bbb E_2(s^{H_2}) &=\Bbb E_2(s^{H_1}s^{\tilde H_0};H_1<\infty)\cr &=\Bbb E_2(s^{H_1};H_1<\infty)\Bbb E_1(s^{H_0})\cr &=\Bbb E_2(s^{H_1})\Bbb E_1(s^{H_0})\cr &=\phi(s)\phi(s).\cr } $$