# 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.

• Why is $E[X_t^2]$ finite a requirement of wide-sense-stationary but not second order stationary? Can you provide a source for this constraint?
– Eric
Oct 12 '16 at 11:17
• Stationary to order 2 says nothing about the moments of the random variables, only about the distributions whereas wide-sense-stationarity is all about the moments and does not require any special properties of the distributions. The most commonly accepted definition of wide-sense-stationarity includes the requirement of finite second moment but if you don't like it, you can discard the requirement and try to persuade others to accept your broader definition as the commonly accepted definition. Oct 12 '16 at 14:23
• I ask because Metrics' comment below disagrees with you here. So your definition of a WSS process is a subset of "random processes of order 2 that are stationary to order 2"?
– Eric
Oct 12 '16 at 20:10
• No, a second-order process (aka finite second moment) that is stationary to order 2 (or more) is a WSS process but stationarity to order 2 is not necessary for a finite-second-moment process to be a WSS process. In other words, my definition of WSS processes includes stationary-to-order-2 processes that have finite second moment. Oct 12 '16 at 21:46

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

• Thanks! Is second order stationarity same as wide-sense stationarity?
– Tim
Jul 24 '13 at 2:00
• Yes @Tim. You can check that on wiki too. Jul 24 '13 at 2:05
• Surprising ...Wiki has separate definitions for weak and second order but there is no reference for second order stationary. Jul 24 '13 at 2:08

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.