I am reading a textbook on Markov chain. The textbook defined the stopping time as follows:
A random variable $T: \Omega→\{0,1,2...\}\cup\{\infty\}$ is called a stopping time for chain $X$ if, for all $n\ge0$, the event $\{T=n\}$ is given in terms of $\{X_0,X_1,...X_n\}$ only.
Then the textbook provides a classical example of hitting time as stopping time. If $A\subseteq S$, and consider the hitting time $H^A$ given by $\{H^A=n\}=\{X_n\in A\}\cap(\cap_{0\le m \lt n} \{X_m\notin A\})$. I understand this example as well.
Then the book proceeds to claim that $\{T=H^A+1\}$ is a stopping time while $\{T=H^A-1\}$ is not.
The last part is where I don't understand. Since $H^A$ is a random variable. Either $T=H^A+1$ or $T=H^A-1$ only shifts the support of the random variable. Shouldn't it still be a stopping time unless stopping time is strictly defined on non-negative integers?
$\{H^{A}-1=n\}=\{H^{A}=n+1\}=\{X_1 \notin A, ..., X_n \notin A, X_{n+1} \in A\}$. This depends on $X_1,X_2,...,X_{n+1}$ and hence it is not a stopping time.