Let $(X_n)_{n\in\mathbb{N}}$ be a Markovchain. How can I then show following equation for all $ n \in \mathbb{N}$,
$ \displaystyle\bigcup_{k=1}^n \lbrace X_k = j \rbrace = \biguplus_{k=1}^n \lbrace X_1 \neq j, \dots, X_{k-1} \neq j, X_k =j \rbrace$ ?
Best regards!
This doesn't have anything to do with Markov chains; it is a matter of set operations. If $$\omega\in\bigcup_{k=1}^n\{X_k=j\},$$ let $k^\star$ be the smallest positive integer such that $\omega\in\{X_{k^\star}=j\}$. Then $$\omega\in\bigcap_{k=1}^{k^\star-1}\{X_k\ne j\}\cap\{X_{k^\star}=j\}\subset\bigcup_{k^\star=1}^n\bigcap_{k=1}^{k^\star-1}\{X_k\ne j\}\cap\{X_{k^\star}=j\}. $$ Conversely, if $$\omega\in\bigcup_{k^\star=1}^n\bigcap_{k=1}^{k^\star-1}\{X_k\ne j\}\cap\{X_{k^\star}=j\} $$ then $$\omega\in \{X_{k^\star}=j\}\subset\bigcup_{k=1}^n\{X_k=j\}. $$