Weak and strict stationarity

A process indexed by integers is strictly stationary if shifting all times in any finite observation vector by the same integer leaves its joint distribution unchanged. Equal one-time marginals alone are insufficient, since the dependence between successive observations could change. An iid sequence is strictly stationary because its joint laws are products of one fixed marginal law. Strict stationarity does not itself impose integrability: an iid Cauchy sequence satisfies the definition despite lacking a finite mean and variance.

Weak stationarity instead requires a finite second moment at every time, one constant mean, and covariance depending only on time difference. We use gamma(h)=Cov(X at t+h, X at t). For a real scalar series, gamma is even. Weak stationarity permits covariance-based methods without assuming the whole process law is Gaussian. It is a statement about the population process, not a claim that sample means or sample covariances computed from two finite stretches will match exactly.

Strict stationarity with finite second moments implies weak stationarity by applying the distributional equality to one- and two-dimensional vectors. The reverse implication fails in general. However, if every finite vector is jointly Gaussian, its law is determined by its mean vector and covariance matrix. Weak stationarity then makes all those Gaussian laws shift invariant, proving strict stationarity. Saying only that each individual observation is marginally Gaussian does not supply that joint-Gaussian assumption.

White noise is another term requiring care. Weak white noise has mean zero, a constant finite variance and zero covariance at every nonzero lag. It need not have independent observations. Jointly Gaussian white noise is independent because its finite covariance matrices are diagonal, but a non-Gaussian process can conceal nonlinear dependence behind zero correlations. Later innovations will initially be defined by orthogonality, so independence must be justified separately when a likelihood calculation needs it.

A time-homogeneous Markov transition law can generate a nonstationary process when its initial distribution differs from an invariant distribution. The transition probabilities stay fixed while marginal probabilities move toward equilibrium. Similarly, a stable autoregression started from a deterministic value has a transient variance before reaching its stationary law. A stationary representation extending into the infinite past and a finite recursion with a chosen initial value are related constructions, but they are not automatically the same process.

A deterministic trend added to a stationary error changes the mean with calendar time. Removing a known trend can reveal a stationary residual; estimating that trend introduces uncertainty and a data-availability question. A rolling plot that looks stable does not prove strict stationarity, ergodicity or independence. Those properties control different aspects of inference, and a structural change can make stationarity a useful local approximation rather than an exact global assumption.