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!


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.

Source : Link , Question Author : Tim , Answer Author : Community

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