user366312
Gold Member
 66
 2
 Summary

Kindly provide me examples of a limiting distribution, and a stationary distribution.
I am totally confused with these two terms.
My Solution:Consider a Markov chain ##(X_n)_n## on ##S=\{1, 2\}## with initial distribution ##α## and the transition matrix
##P =
\begin{bmatrix}
2/3 & 1/3 \\
2/3 & 1/3 \\
\end{bmatrix}##
1. Limiting distribution = ?
2. Stationary distribution = ?
##\underline {\text{Limiting Distribution}}##
##
P^2 = \begin{bmatrix}
2/3 & 1/3 \\
2/3 & 1/3 \\
\end{bmatrix}
\begin{bmatrix}
2/3 & 1/3 \\
2/3 & 1/3 \\
\end{bmatrix} =
\begin{bmatrix}
2/3 & 1/3 \\
2/3 & 1/3 \\
\end{bmatrix}
##
So, the limiting distribution of ##P## is ##P## itself.
____
##\underline {\text{Stationary Distribution}}##
Let the stationary distribution ##\pi = \begin{bmatrix} p & 1p \end{bmatrix}##.
So,
##\pi P = \pi ##
##\Rightarrow \pi (P1) = 0##
##\Rightarrow \begin{bmatrix} p & 1p \end{bmatrix} \left(\begin{bmatrix}
2/3 & 1/3 \\
2/3 & 1/3 \\
\end{bmatrix}  \begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}\right) = 0##
##\Rightarrow \begin{bmatrix} p & 1p \end{bmatrix} \begin{bmatrix}
1/3 & 1/3 \\
2/3 & 2/3 \\
\end{bmatrix} = 0##
##\Rightarrow \begin{bmatrix} \frac{p}{3}+\frac{2}{3}+\frac{2p}{3} & \frac{p}{3} + \frac{2}{3} + \frac{2p}{3} \end{bmatrix} = 0##
##\Rightarrow p = 2/3##
So,
##\pi = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} \end{bmatrix}##
___
 Are the terms Limiting distribution and Stationary distribution properly perceived here?
Last edited: