Examples from Markov systems

A three-state stock market

Order the states as bull, bear and stagnant. The transition matrix is

For example, the first column says that a bull market is followed by a bull market with probability $0.65$, a bear market with probability $0.10$ and a stagnant market with probability $0.25$. Each column sums to $1$ and every entry is non-negative, so this is a transition matrix.

If the market starts in the bear state, then

The stock market in the long run

For this transition matrix, the eigenvalues and corresponding eigenvectors are

These eigenvectors form a basis, $1$ has multiplicity $1$ and

For the initial bear state,

The last two modes vanish as $k\to\infty$, leaving a multiple of $v_1$. Its entries sum to $229+70+109=408$, so the limiting probability vector is

The entries are non-negative and sum to $1$ and $A\mathbf{p}=\mathbf{p}$ because $Av_1=v_1$. Thus $\mathbf{p}$ is the stationary distribution: the long-run probabilities of bull, bear and stagnant states.

The other two eigenmodes are temporary departures from this distribution. Both shrink and the negative eigenvalue makes its departure alternate in sign. The initial bear state therefore affects the short run but is forgotten in the long run.