I am new to cross validated so I hope my question belongs here. I saw in a paper where I study someone claiming the following:
Given a $ \{ X_n \}_{n=0}^{\infty} $ be a homogeneous Markov chain (the transition probabilities are independent of the time) with a probability matrix P. We have a specific state $ j $ and we know $ P(X_0=j)=1 $. It is claimed the following holds:
$ P(X_{n+i}=j| X_{n+i-1} \neq j ... X_{n+1} \neq j , X_n=j) =\sum_{l \neq j} P(X_{n+i}=j| X_{n+i-1} = l \space , \space X_{n+i-2} \neq j \dots X_{n+1} \neq j \space , \space X_n=j) $
My thoughts were that this is generally not true as the conditional law of total probability each term has to be multiplied by $ P(X_{n+i-1}=l|X_{n+i-1} \neq j ,X_{n+i-2}\neq j ... X_n=j)$ but maybe due to Markov property it is different or I have a mistake? I just wanted to ask the community and I thank all kind helpers.