I am taking a course about stochastic processes and I have a question about the hitting time of Markov chains. Any thoughts, explanations are welcome. For completeness, let me give the definition of hitting time.
$$H^A : \Omega \rightarrow \mathbb{N}_0 \cup \{\infty\}$$ $$w \rightarrow \inf\{n \in \mathbb{N}_0 \cup \{\infty\} \colon X_n(w) \in A \}$$
It states in the book that $H^A$ is a random variable. (http://www.statslab.cam.ac.uk/~james/Markov/s13.pdf) Why is it defined as a random variable in the first place ? I do not understand the relationship between the probability space and the hitting time. I understand the mathematical intuition behind the definition but I want to understand the details. I am fairly new at probability theory, so you can also suggest references about the topic.
Thanks a lot for reading !
(I found the answer to my question and I will post it here as an answer, in case someone also has problems with the definitions.)
The sample space (i.e. $\Omega$ in $X_i : \Omega\to E$) associated with the random variables ($X_i$) of the Markov chain consists of sequences of states. For $w \in \Omega$, $X_i(w)$ is the state at the $i^{\text{th}}$ coordinate of the sequence $w$.
Now, $H^A$ is defined as a random variable because it makes use of the random variables $X_i$. For every $w \in \Omega$, $H^A$ is found by $$ \inf\{n \in \mathbb{N}_0 \cup \{\infty\} \colon X_n(w) \in A \} $$ that is, looking at the sequence $w$ and finding the infimum of $n$'s that satisfy the above condition. Since $H^A$ is a random variable, probabilities like $P(H^A(w) \lt \infty)$ can be found in the same way for random variables.