4 - Qualitative Behavior of Chains#
If we want to find \(\lim_{n \rightarrow \infty} q P^m\), there are some special cases where this is easy to identify.
4.1 - Two Special Matrices that Can Be Computed by Hand#
Case 1#
Assume we have the following one-step transition probability matrix:
\[\begin{split} \begin{bmatrix}
0 & 1 & 0 \\
1 - p & 0 & p \\
0 & 1 & 0
\end{bmatrix} \end{split}\]
We observe through matrix multiplication that \(P = P^3\) and
\[\begin{split} P^2 = P^4 = \begin{bmatrix}
1 - p & 0 & p \\
0 & 1 & 0 \\
1 - p & 0 & p
\end{bmatrix} \end{split}\]
Thus, we have \(P^n = P\) when \(P\) is odd, and \(P^n = P^2\) when \(P\) is even.
Case 2#
Assume the one-step transition probability matrix is
\[\begin{split} P = \begin{bmatrix}
0.6 & 0 & 0.4 \\
0 & 1 & 0 \\
0.6 & 0 & 0.4
\end{bmatrix} \end{split}\]
We can see that \(P = P^2 = \dots = P^n\) where \(n\) is an integer.
4.2 - Long Term and Short Term Behavior of the Chain#
It is difficult to determine the long-term behavior of a chain based on the matrix computation of \(q_0 \times P^n\).