What is a second order stationary process?

Question

I was wondering how his "second-order stationary process" is defined in Brockwell and Davis' Introduction to Time Series and Forecasting:

The class of linear time series models, which includes the class of autoregressive moving-average (ARMA) models, provides a general framework for studying stationary processes. In fact, every second-order stationary process is either a linear process or can be transformed to a linear process by subtracting a deterministic com- ponent. This result is known as Wold's decomposition and is discussed in Section 2.6.

In Wikipedia,

The case of second-order stationarity arises when the requirements of strict stationarity are only applied to pairs of random variables from the time-series.

But I think the book has a different definition from Wikipedia's, because the book uses stationarity short for wide-sense stationarity, while Wikipedia uses stationarity short for strict stationarity.

Thanks and regards!

Answer 1

There can be some confusion of terms here depending on whether the adjective seond-order is considered to be modifying stationary or random process (or both!). To some people,

• A second-order random process $\{X_t \colon t \in \mathbb T\}$ is one for which $E[X_t^2]$ is finite (indeed bounded) for all $t \in \mathbb T$. For us electrical engineers who apply (or mis-apply!) random process models in studying electrical signals, $E[X_t^2]$ is a measure of the average power delivered at time $t$ by a stochastic signal, and so all physically observable signals are modeled as second-order processes. Note that stationarity has not been mentioned at all and these second-order processes might or might not be stationary.

• A random process that is stationary to order $2$, which we can (but perhaps should not) call a second-order stationary random process provided we agree that second-order modifies stationary and not random process, is one for which $\mathbb T$ is a set of real numbers that is closed under addition, and the joint distribution of the random variables $X_t$ and $X_{t+\tau}$ (where $t, \tau \in \mathbb T)$ depends on $\tau$ but not on $t$. As the link provided by AO shows, a random process stationary to order $2$ need not be strictly stationary. Nor is such a process necessarily wide-sense-stationary because there is no guarantee that $E[X_t^2]$ is finite: consider for example a strictly stationary process in which the the $X_t$'s are independent Cauchy random variables.

• A second-order random process (meaning finite power as in the first item above) that is stationary to at least order $2$ is wide-sense-stationary.

OK, so that is the perspective from a different set of users of random process theory. For more details, see, for example, this answer of mine on dsp.SE.

Answer 2

Second order stationary is a weak stationary or covariance stationary. See the following excerpt from Time Series Analysis, J. Hamilton (1994) p. 108

Answer 3

I'm guessing its the same as "weakly stationary". That means that $(x_k,\dots,x_{k-l})$ all (for all $k$, and any $l)$ have the same expectation and covariance matrix but not necessarily the same distribution.