First property of discrete time homogenous markov chain

Question

I'm trying to understand the properties of a DTHMC. I am having trouble understanding with the first one. My textbook says -

"$X_t$ takes values in $X$ for all $t$ (i.e. $X_t$ is a random variable with values in $X$)"

can someone explain that?

Answer 1

Hmmm ... it seems that your book is confusingly using $X$ to denote both the process and the state space. Let me change the state space to $S$ to make things simpler.

This is the description of a stochastic process; it is just a collection of random variables. Instead of just one or two, you have a random variable for every $t$ in some index set $T$: $$ X_t : (\Omega,\mathcal{F},P) \to (S,\mathcal{S}) $$ (here $(\Omega,\mathcal{F},P)$ is your probability space, and $\mathcal{S}$ is some $\sigma$-fields on your state space $S$).

You can think of $t$ as corresponding to time in many cases, so $X_t$ might be the position at time $t$ of a particle which moves around randomly in some state space.