Proving identity of Markov chain.

Question

I am in a need of proving the following identity. Hope someone can help me. Let $\{X_n, n\geqslant0\}$ be a homogeneous Markov Chain. Show that $$P(X_{n+1}=k_1,...,X_{n+m}=k_m \mid X_0=i_0,...,X_n=i)=P(X_1=k_1,...,X_m=k_m \mid X_0=i) $$I know I should use multiple conditioning and the Markov property in some way, but can't figure out how. Thanks in advance.

Answer 1

The Markov property yields \begin{align} &\mathbb P(X_{n+1}=k_1,\ldots, X_{n+m}=k_m\mid X_0=i_0,\ldots,X_n=i)\\ =& \mathbb P(X_{n+1}=k_1,\ldots, X_{n+m}=k_m\mid X_n=i).\end{align} Time homogeneity yields \begin{align} &\mathbb P(X_{n+1}=k_1,\ldots, X_{n+m}=k_m\mid X_n=i)\\ =& \mathbb P(X_1=k_1,\ldots, X_m=k_m\mid X_0=i). \end{align}