Stationary distribution may refer to: * and , a special distribution for a

Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...

such that if the chain starts with its stationary distribution, the marginal distribution of all states at any time will always be the stationary distribution. Assuming irreducibility, the stationary distribution is always unique if it exists, and its existence can be implied by positive recurrence of all states. The stationary distribution has the interpretation of the limiting distribution when the chain is irreducible and aperiodic. * The

marginal distribution In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variable ...

of a stationary process or stationary

time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...

* The set of

joint probability distribution A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...

s of a stationary process or stationary

In some fields of application, the term stable distribution is used for the equivalent of a stationary (marginal) distribution, although in

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

the term has a rather different meaning: see

stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be st ...

. 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

eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...

s for which the

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

is unity.

See also