= Weakly stationary process
= Weak stationarity
{synonym}
A real <stochastic process> $(X_t)_{t\in\mathbb Z}$ is a <weakly stationary process> if it has finite <second moments>, a constant <expected value> $\mu$, and an <autocovariance> depending only on the <lag of a time series>:
$$
\mathbb E[X_t]=\mu,\qquad \mathbb E[X_t^2]<\infty,\qquad
\operatorname{Cov}(X_{t+h},X_t)=\gamma(h)\quad\text{for all }t,h\in\mathbb Z.
$$
In particular its <variance> is the same at every time, $\operatorname{Var}(X_t)=\gamma(0)$. This is often called second-order stationarity. It makes observations from different parts of a record comparable through the same <autocorrelation function>. Bartlett uses this second-order meaning of stationarity in https://www.stat.berkeley.edu/~bartlett/courses/153-fall2010/lectures/2.pdf[Introduction to Time Series Analysis, lecture 2: mean, covariance, and stationarity].
For a worked contrast, let $\varepsilon_t$ be centered <white noise> of <variance> $\sigma^2>0$. The process $X_t=5+\varepsilon_t$ has <mean> $5$, <variance> $\sigma^2$, and zero <autocovariance> at nonzero lags, so it is weakly stationary. The process $Y_t=t+\varepsilon_t$ has exactly the same <covariance> structure, but its <mean> is $t$, so it is not weakly stationary. Constant <variance> alone is insufficient.
Weak stationarity need not imply <strict stationarity>. For example, choose <independent random variables> at all integer times, using a standard <normal distribution> at even times and a <Rademacher distribution> at odd times. Every coordinate has zero <mean> and unit <variance>, and distinct coordinates have zero <covariance>. Nevertheless a shift by one changes the marginal <probability distribution> from continuous to two-point. This also illustrates why <weak white noise> need not be identically distributed.
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