Spectral view for two states
Section of Discrete-time Markov chains, part of Stochastic processes
Spectral view for two states
| Type | Section |
| Taught in | Discrete-time Markov chains |
| Part | Stochastic processes |
Let
P=\begin{pmatrix} 1-a & a \\ b & 1-b \end{pmatrix}, \qquad a,b>0.
The eigenvalues are 1 and 1-a-b. The stationary law is
\pi=\Bigl(\frac{b}{a+b},\frac{a}{a+b}\Bigr).
The second eigenvalue controls the decay of a deviation from \pi. Write a starting row as \pi plus a multiple of the left eigenvector for 1-a-b. After n steps that deviation is multiplied by (1-a-b)^n. Whenever |1-a-b|<1, which holds as soon as a and b are not both 0 or both 1 in the alternating way below, P^n approaches the matrix with every row equal to \pi.
At a=b=1 the second eigenvalue is -1. The chain swaps states at every step, P^n alternates between P and I, and the powers have no limit. A stationary distribution still exists: \pi=(1/2,1/2). This is the same two-state alternation as the periodic example on Markov chains. The eigenvalue makes the oscillation visible: magnitude 1, and not equal to the eigenvalue 1.
Used in
See also
Discrete-time Markov chain · Stationary distribution · Transition matrix · Hitting probabilities · Markov assumption as a chosen state · Stationarity and the law from one start · Doubly stochastic matrix