What is a second order stationary process? 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!
 A: 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.
A: Second order stationary is a weak stationary or covariance stationary. See the following excerpt from Time Series Analysis, J. Hamilton (1994) p. 108

A: 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.
