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I am reading my lecture notes on Markov chains, and in the proof of one proposition the following statement is made:

"For $n = 1,2, \dots$ the event $\{n \leq T_k\}$ depends only on $X_0, \dots, X_{n-1}$". Where $T_k$ is the first passage time on $k$. I'm slightly confused about this, because it seems to me that the event also depends on the value of $X_n$. Can someone please point out what I'm missing?

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up vote 2 down vote accepted

Indeed the event mean the first hitting time is greater than or equal to $n$ so either $X_n=k$ or not is still the same event so it is independent to the event.

$X_n\ne k,X_{n+r}=k$ and $X_n=k$ are also an event in $\{n\le T_k\}$ so obviously the event doesn't depends on $X_n$ but for $0,...,n-1$ it does depend as if any $r\in [0,...,n-1] s.t. X_r=k$ the the event would become $\{r\le T_k\}$

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$$[T_k\geqslant n]=\bigcap_{i=0}^{n-1}[X_i\ne k]\implies[T_k\geqslant n]\in\sigma((X_i)_{0\leqslant i\leqslant n-1})$$

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