nLab stationary stochastic process

Idea

A stochastic process is called stationary if the probability of each trajectory is invariant under (rigid) time translation?.

It is a way to encode mathematically a system which might exhibit randomness? as well as memory?, but which does not explicitly depend on time.

Definition

A stochastic process $(X_t)_{t\in T}$ (for $T=\mathbb{N},\mathbb{R}$, etc.) is called stationary if and only if for every finite marginal $(X_{t_1},\dots,X_{t_n})$ and each $s\in T$, the tuples

$$(X_{t_1},\dots,X_{t_n}) \qquad and \qquad (X_{t_1+s},\dots,X_{t_n+s})$$

have the same joint distribution.

Note that this condition is stronger than requiring that the distributions of the single $X_t$ are time-invariant: also the correlations, of any order, are time-invariant, as long as all the times are translated “rigidly”, by the same amount $s$.

Equivalently, it is stationary if the complete joint distribution on $X^T$ is shift-invariant, i.e. it is a measure-preserving dynamical system.

Examples

• A Markov chain is stationary if and only if its transition kernel is a time-independent measure-preserving Markov kernel.
• White noise is a continuous-time stationary process.
• Every exchangeable process is stationary, in particular, Bernoulli processes.
• Every ergodic process is stationary.
• Every time-reversible process is stationary.

Related entries

• probability theory, categorical probability
• invariant measure, invariant set, equilibrium
• ergodic decomposition theorem, ergodic theorem?
• de Finetti's theorem, zero-one law
• Markov process?, Markov kernel, Markov category
• shifts, Bernoulli process

References

• Wikipedia, Stationary process

category: probability