I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution
Let $t_i$ denote the return time of a state $i$ of a Markov chain, then we have
$$ \{t_i \le n\} = \bigcup_{l=1}^{n} \{X_l=i,X_{l+1} \neq i,...,X_n \neq i\}.$$
Furthermore, my textbook says that this union is disjoint.
I don't get it: If a return time is less than a number $n$, then it can be either $1,2,...,n$. Now the point of the return time is, that if it is let's say $3$, then $X_1 \neq i$ and $X_2\neq i$, but $X_3=i$ again. Furthermore, we should not care about what $X_4,...,X_n $ are. So, I don't see what this has to do with the union there. Could anybody help me with that please?
The event in the bracket on the righthand side says:
the last time $X$ hit $i$ is $l$, for $l\leq n$
When you take union over $l$, of course this is the event that $X$ hits $i$ before $n$.
It is a weird way of writing but it holds. You are writing the union over the last hitting of $i$ before $n$ (assuming $X$ hit $i$)