What is "stationary distribution"

a. (context mathematics stochastic processes of a Markov chain English) a row vector $pi$ whose entries sum to $1$ that satisfies the equation $pi\; P=pi$, where $P$ is the transition matrix of the Markov chain.

Stationary distribution may refer to:

• The limiting distribution in a Markov chain
• The marginal distribution of a stationary process or stationary time series
• The set of joint probability distributions of a stationary process or stationary time series

In some fields of application, the term stable distribution is used for the equivalent of a stationary (marginal) distribution, although in probability and statistics the term has a rather different meaning: see stable distribution.

Crudely stated, all of the above are specific cases of a common general concept. A stationary distribution is a specific entity which is unchanged by the effect of some matrix or operator: it need not be unique. Thus stationary distributions are related to eigenvectors for which the eigenvalue is unity.