I was wondering how his “secondorder 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
movingaverage (ARMA) models, provides a general framework for studying stationary processes. In fact, every secondorder 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 secondorder stationarity arises when the requirements of strict stationarity are only applied to pairs of random variables from the timeseries.
But I think the book has a different definition from Wikipedia’s, because the book uses stationarity short for widesense stationarity, while Wikipedia uses stationarity short for strict stationarity.
Thanks and regards!
Answer
There can be some confusion of terms here depending on whether the adjective seondorder
is considered to be modifying stationary or random process (or both!).
To some people,

A secondorder 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 misapply!) 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 secondorder
processes. Note that stationarity has not been mentioned at all
and these secondorder 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 secondorder stationary random process provided we agree that secondorder 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 widesensestationary 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 secondorder random process (meaning finite power as in the first item above)
that is stationary to at least order $2$ is widesensestationary.
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.
Attribution
Source : Link , Question Author : Tim , Answer Author : Community