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Stationary

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Definition
InfoTheoryStochasticProcesses

The source {Xi}i=1∞\{X_i\}_{i=1}^{\infty} is called stationary if P(X1=a1,⋯ ,Xn=an)=P(X1+τ=a1,⋯ ,Xn+τ=an)P(X_1=a_1,\cdots,X_{n}=a_n)=P(X_{1+\tau}=a_1,\cdots,X_{n+\tau}=a_n) ∀an=(a1,⋯ ,an)∈Xn\forall a^n=(a_1,\cdots,a_n)\in\mathcal{X}^n and integers n,τ≥1n,\tau\ge1.

This would imply that the joint distribution is invariant under time shifts.

If source {Xi}i=1∞\{X_i\}_{i=1}^{\infty} is stationary, then it is identically distributed or P(Xn=a)=P(Xn+τ=a) ∀a∈X\mboxandn,τ≥1P(X_n=a)=P(X_{n+\tau}=a) \ \forall a\in\mathcal{X} \mbox{ and } n,\tau\ge 1

A DMS source is stationary.

If a time-invariant MC is identically distributed, then it is a stationary process.

If a time-invariant MC {Xi}\{X_{i}\} has its initial distribution PX1P_{X_1} given by the chain’s stationary distribution Π\Pi (i.e. PX1=ΠP_{X_1}=\Pi), then the MC is a stationary process.