A Conceptual Vocalubary of Time Series Analysis

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In time series analysis context, if the time series is one for which the probabilistic behavior of every collection of values $$ \{x_{t_1},x_{t{_2},...,x_{t_k}}\} $$is identical to that of the time shifted series$$\{x_{t_{1+h}},x_{t_{2+h}},...,x_{t_{k+h}}\}$$for all \( k=1,2,... \), all time points \(t_1,t_2,...,t_k\) and all time shifts \( h=0, \pm1, \pm2,... \), time series is said to be a strictly stationary.

It is difficult to assess strict stationarity from data. Rather than imposing conditions on all possible distributions of a time series,  a milder version called weakly stationary is used that imposes conditions only on the first two moments of the series such as a) the mean value function \( \mu_t \) is constant and does not depend on time \(t\) and b) the autocovariance function depends  on \(s\) and \(t\) only through their difference \(|s-t|\).

Stationarity requires regularity in the mean and autocorrelation functions so that these quantities (at least) may be estimated by averaging. It should be clear that a strictly stationary, finite variance, time series is also stationary. The converse is not true in general. One important case where stationarity implies strict stationarity is if the time series is Gaussian [meaning all finite collections of the series are Gaussian].