Let $X_{n}$ be a MC s.t. $(X_n)_{n\ge0}\sim \mbox{Markov}(\lambda,P)$, and $m\ge0$ be a fixed integer. Then conditional on $\{X_m=i\}$ 1. $(X_{m+n})_{n\ge0}$ is independent of $X_0,\cdots,X_{m-1}$ 2. $(X_{m+n})_{n\ge0}$ is $\mbox{Markov}(\delta_i,P)$

where $\delta_{i}$ is the Dirac Distribution at state $i$.

Let $\lambda$ be a distribution on $S$, and $P$ be a stochastic matrix. Assume for any $N\ge0$, and any states $i_0,\cdots,i_N\in S$ $$P(X_0=i_0,\cdots,X_N=i_N)=\lambda_{i_0}p_{i_0i_1}\cdots p_{i_{N-1}i_N}$$ Then $X_n,n\ge0\sim Markov(\lambda,P)$, or $X_{n},n\ge0$ is a MC.

Let $(X_{n})_{n\ge0}\sim\mbox{Markov}(\lambda,P)$ be a time homogeneous MC, and $T$ be a stopping time. Then for any integer $m\ge0$ and state $i\in S$, conditional on $\{T=m,X_{T}=i\}$, 1. $(X_{T+n})_{n\ge0}$ is conditionally independent of $X_{0},\cdots,X_{T},T$ 2. $(X_{T+n})_{n\ge0}$ is $\mbox{Markov}(\delta_{i},P)$

where $\delta_{i}$ is the Dirac Distribution at state $i$.

Let $(X_{n})_{n\ge0}\sim\mbox{Markov}(\lambda,P)$ be a time homogeneous MC, and $T$ be a stopping time such that $P(T<\infty)=1$. Then for any state $i\in S$, conditional on $X_{T}=i$, 1. $(X_{T+n})_{n\ge0}$ is conditionally independent of $X_{0},\cdots,X_{T},T$ 2. $(X_{T+n})_{n\ge0}$ is $\mbox{Markov}(\delta_{i},P)$

where $\delta_{i}$ is the Dirac Distribution at state $i$.

If $\{X_{t}:t\ge0\}\sim\mbox{Markov}(\lambda,Q)$, then fix a time $s\ge0$, conditional on $\{X_{s}=i\}$, we have that - $\{X_{t+s}:t\ge0\}\sim\mbox{Markov}(\delta_{i},Q)$ - $\{X_{t+s}:t\ge0\}$ is conditionally independent of $\{X_{r}:t< s\}$