15 - Joint Distribution of a Markov Chain

Joint Distribution of a Markov Chain

Let $X=(X_0,X_1,X_2,\cdots )$ be a Markov Chain with values in $I$

For $i_0,i_1,\cdots ,i_n\in I$ then

$$\mathbb{P}(X_0=i_0,X_1=i_1,\cdots ,X_n=i_n)={\lambda }_{i_0}{p}_{i_0{i}_1}{p}_{i_1{i}_2}\cdots p_{i_{n-1}{i}_n}$$

Proof

$$\begin{array}{rl}\mathbb{P}(& X_0=i_0,X_1=i_1,\cdots ,X_n=i_n)\\ & =\mathbb{P}(X_0=i_0)\mathbb{P}(X_1=i_1\mid X_0=i_0)\mathbb{P}(X_2=i_2\mid X_1=i_1,X_0=i_0)\times \cdots \\ & \cdots \times \mathbb{P}(X_n=i_n\mid X_{n-1}=i_{n-1},\cdots ,X_0=i_0)\\ & =\mathbb{P}(X_0=i_0)\mathbb{P}(X_1=i_1\mid X_0=i_0)\mathbb{P}(X_2=i_2\mid X_1=i_1)\cdots \mathbb{P}(X_n=i_n\mid X_{n-1}=i_{n-1})\\ & ={\lambda }_{i_0}{p}_{i_0{i}_1}{p}_{i_1{i}_2}\cdots p_{i_{n-1}{i}_n}\end{array}$$