Markov Chains

A certain town is in a certain country (this is a hypothetical problem). Each year ten percent of the town dwellers move to other parts of the country. Each year one percent of the people from elsewhere move to the town. Assume that there are two states $s_{T}$, living in town, and $s_{C}$, living elsewhere.

• Construct the transition matrix.

• Starting with an initial distribution $s_{T}=0.3$ and $s_{C}=0.7$, get the results fo…

Above we define a transition matrix to have each entry nonnegative and each column sum to $1$.

• Check that the three transition matrices shown in this Topic meet these two conditions. Must any transition matrix do so?

• Observe that if $A\vec{v}_{0}=\vec{v}_{1}$ and $A\vec{v}_{1}=\vec{v}_{2}$ then $A^{2}$ is a transition matrix from $\vec{v}_{0}$ to $\vec{v}_{2}$. Show that a power of a transition matrix is also a transition matrix.

• Generalize the prior item by proving that…