Weak stationarity
Topic of Dependence along a series, part of Time series
Weak stationarity
| Type | Topic |
| Taught in | Dependence along a series |
| Part | Time series |
| Questions using it | 2 |
A series is weakly, or second-order, stationary when its mean is constant and \operatorname{Cov}(X_s,X_t) depends only on t-s. A Gaussian series that is weakly stationary is strictly stationary.
In Dependence along a series
Without some repetition, a series of length T does not identify a different mean at every time and a different covariance for every pair. Strict stationarity says that any block of observations has the same joint distribution as the same block shifted by \tau. The path changes. The distribution does not. Weak stationarity asks only for two moments: a mean \mu that does not depend on time, and a covariance that depends on the two times only through their difference, \gamma_{t-s}=\mathrm{Cov}(X_s,X_t). The lag-zero value \gamma_0=\mathrm{Var}(X_t) is then constant as well. The autocorrelation is \gamma_h/\gamma_0, which lies in [-1,1]. For a Gaussian process, weak stationarity and strict stationarity are the same statement. Otherwise weak is weaker.
Used in
- Taught in Dependence along a series
- Questions that use it (2) — C43 · C45 (answer)
- All 7 questions on Dependence along a series
See also
Stationarity of a time series · Strict stationarity · Autocorrelation function · Variogram · Wold decomposition · Autoregression and moving averages · Random walk with drift · White noise