1. Covariance Stationary Stochastic Processes

A stochastic process is a collection of random variables indexed by a variable \(t\) belonging to a set \(T\). For a continuous time stochastic process, \(T\) is \(\mathbb{R}\), the set of real numbers, or an interval in \(\mathbb{R}\). It is imagined that there is an underlying probability space, which is characterized by a 3-tuple \((\Omega, \mathcal{F}, P)\), where \(\Omega\) is a collection of points, \(\mathcal{F}\) a collection of sets in \(\Omega\), and \(P\) a probability measure. The sets in \(\mathcal{F}\) are collections of points for which the probability can be computed with the probability measure \(P\). For example, for a set \(A \in \mathcal{F}\), \(P(A)\) is a real number in the interval \([0, 1]\). A set \(A \in \mathcal{F}\) is called an event. A random variable is a “measurable function” of \(w \in \Omega\). This means that a random variable \(x(w)\) is a function from \(\Omega\) to the real line for which the probability is well defined for all events of the form \(A = [w : x(w) \in [a, b]]\) for all intervals with \(a \leq b\).

A stochastic process \(x_t\) is a collection of such measurable functions \(x_t = x(t,w)\), for \(t \in T\). Hereafter, we set \(T = \mathbb{R}\), so that we are studying a continuous time stochastic process. We begin by assuming that \(x(t,w)\) is a scalar random variable, i.e., \(x(t,w)\) is a “univariate stochastic process.” In this construction, for each “drawing” of \(w \in \Omega\), one draws an entire function of time \(x(t,w)\). Each such drawing of a function of time \(x_t = x(t,w)\) is called a “realization of the stochastic process,” or a “sample path of the stochastic process.” We might desire to characterize the average behavior of these functions \(x_t = x(t,w)\) both across realizations or \(w\)’s, and across time for a given single realization or drawing of \(w\). For example, corresponding to the “mean”, we could compute \(\int x(t,w)\,dP(w)\), which is the average across realizations, at a given time \(t\); alternatively, we could compute \(\lim_{T \to \infty} (2T)^{-1} \int^T_{-T} x(t,w)\,dt\), the average across \(t\) during a single realization \(w\). In these pages we spend most of our time characterizing the behavior of the stochastic process \(x_t = x(t,w)\) across realizations \(w\). Under some regularity conditions on the stochastic process \(x(t,w)\), the behavior of averages across \(w\)’s is closely reflected in behavior of averages across time for a single \(w\). These regularity conditions are conditions for “ergodicity,” and must be imposed to acquire a practical theory of estimation. For now, it is important to realize that averages across time within a single realization are logically distinct from averages over realizations at a point in time.

Note

Every estimator in the second part of this book is a time average from a single realization, asked to converge to an ensemble quantity: the sampled autocovariances of 21. Inferring a Continuous-Time System from Discrete-Time Data: An Appreciation of A. W. Phillips (1959), the matrix covariogram of 20. Aggregation Over Time and the Inverse Optimal Predictor Problem for Adaptive Expectations in Continuous Time, the discrete spectral density of 17. Discrete Sampling: The Folding Formula. Two conditions deliver that convergence, and they differ in depth.

Mean square ergodicity asks that the sample mean converge to \(\mu\). It restricts second moments only, and the processes of this book satisfy it. 10. The Cramér Representation shows that it holds precisely when the spectral distribution puts no mass at frequency zero, which holds for any purely linearly indeterministic process.

Covariance ergodicity asks that the sample autocovariances converge to \(C(\tau)\). It restricts fourth moments. Nothing in Chapters 1–17 settles it, and two processes with identical \(C(\tau)\) can differ in whether it holds. It requires either Gaussianity or a condition on fourth cumulants.

A1. Ergodicity and the Consistent Estimation of Second Moments develops both, and says what they do and do not buy for estimating spectra and cross-spectra.

We define the following three functions which measure population averages across realizations:

(1)\[\begin{split}\begin{aligned} \mu(t) &= \int x(t,w)\,dP(w) \qquad t \in \mathbb{R} \\ &= E x(t,w) \\[4pt] R(t_1,t_2) &= \int x(t_1,w)\,x(t_2,w)\,dP(w); \ t_1, t_2 \in \mathbb{R} \\ &= E x(t_1,w)\,x(t_2,w) \\ C(t_1,t_2) &= \int \big(x(t_1,w) - \mu(t_1)\big)\,\big(x(t_2,w) - \mu(t_2)\big)\,dP(w), \\ &= E \big(x(t_1,w) - \mu(t_1)\big)\,\big(x(t_2,w) - \mu(t_2)\big)\ \ t_1, t_2 \in \mathbb{R} \end{aligned}\end{split}\]

We assume that the integrals on the right side exist, which means that the stochastic process possesses finite first and second moments. The function \(\mu(t)\) is called the mean function, while \(R(t_1,t_2)\) is called the autocorrelation function, and \(C(t_1,t_2)\) is called the autocovariance function. Often we shall be studying a stochastic process for which \(\mu(t) = 0\), in which case \(C(t_1,t_2) = R(t_1,t_2)\).

Note

Following an older usage common in the engineering literature on which these notes draw, \(R\) is here the second moment function \(E\,x(t_1)x(t_2)\). It is not normalized to lie in \([-1,1]\), and it is not centred unless \(\mu = 0\). Modern statistics reserves “autocorrelation” for the normalized quantity \(C(\tau)/C(0)\). We work almost always with zero-mean processes, where \(R\) and \(C\) coincide, so no harm follows. Read \(R\) as the autocovariance. 8. Spectral Densities calls the same object the covariogram. The spectral density is the Fourier transform of \(R\), not of any normalized version of it.

In the following definition, we introduce the notion of stationarity.

Definition 1

A stochastic process is said to be covariance stationary (or second-order stationary or wide-sense stationary) if

(2)\[\begin{split}\begin{aligned} \mu(t) &= \mu\ \text{ for all } t \in \mathbb{R} \\ R(t_1,t_2) &= R(t_1 + \varepsilon, t_2 + \varepsilon)\ \text{ for all } t_1, t_2 \in \mathbb{R}\ \text{ and all } \varepsilon \text{ in } \mathbb{R} \end{aligned}\end{split}\]

According to (2), the mean \(\mu(t)\) is a constant function of time, \(\mu\), and the covariance between \(x(t_1)\) and \(x(t_2)\) equals the covariance between \(x(t_1 + \varepsilon)\) and \(x(t_2 + \varepsilon)\), so that the covariance between \(x(t)\)’s at two distinct dates depends only on the difference between the two dates. Setting \(\varepsilon = -t_2\), and letting \(t_1 - t_2 = \tau\), we have \(R(t_1, t_2) = R(t_1 - t_2, 0) = R(t_1, t_1 - (t_1 - t_2))\) or

\[ R(t_1, t_1 - \tau) = R(\tau, 0). \]

Adopting an inconsistent but convenient notation, for covariance stationary processes, we shall denote

\[ E x(t)\, x(t - \tau) = R(\tau) \]

where \(R(\tau)\) is understood as \(R(\tau, 0)\). Similarly, we shall also use the notation \(C(\tau, 0) = C(\tau)\).

It is possible to characterize the first and second moments of a covariance stationary stochastic process by its mean \(\mu\) and its autocovariance function \(C(\tau)\). It is useful to know the properties that a function \(C(\tau)\) must have in order to be the autocovariance function of some covariance stationary stochastic process. To be the autocovariance function of a covariance stationary stochastic process, it is necessary and sufficient that \(C(\tau)\) be a positive semidefinite function. There are several equivalent ways to express the concept of positive semidefiniteness of a function. One way is as follows. Let \(x(t)\) be a stochastic process with mean zero and autocovariance function \(C(\tau)\). Then consider forming a random variable

\[ y_h = \int^\infty_{-\infty} h(s)\, x(t - s)\, ds \]

where \(h\) is a function that is a member of the class \(L^2(-\infty, \infty)\) of square integrable functions, i.e., \(\int^\infty_{-\infty} h(s)^2\, ds < +\infty\). Evidently, all such random variables \(y_h\) must have nonnegative variance. This is the requirement that

(3)\[\int^\infty_{-\infty} \int^\infty_{-\infty} h(s)\, h(\tau)\, C(\tau - s)\, ds\, d\tau \geq 0 \ \text{ for all } h(s) \in L^2(-\infty, \infty).\]

Condition (3) is the condition that \(C(\tau)\) is positive semidefinite. That \(C(\tau)\) be positive semidefinite is a necessary and sufficient condition for \(C(\tau)\) to be an autocovariance function for some covariance stationary stochastic process.

Another way to characterize positive semidefiniteness of \(C(\tau)\) is as follows. Let \(t_1, \ldots, t_n\) be any finite set of real numbers. Then use \(C(\tau)\) to form the \((n \times n)\) covariance matrix for the vector \((x(t_1) \ldots x(t_n))\). For every finite \(n\), and every collection \((t_1, \ldots, t_n)\), the covariance matrix must be positive semidefinite if the function \(C(\tau)\) is to be positive semidefinite.

Yet another way to characterize positive semidefiniteness of \(C(\tau)\) can be given using concepts to be introduced in 8. Spectral Densities. It is true that \(C(\tau)\) is positive semidefinite if and only if the function

\[ S(\omega) = \int^\infty_{-\infty} C(\tau)\, e^{-i \omega \tau}\, d\tau \geq 0\ \text{ for all } \omega \in (-\infty, \infty). \]

We now define a more restrictive form of stationarity:

Definition 2

A stochastic process is said to be stationary or strictly stationary if for any finite \(n\), any collection of dates \([t_1, \ldots, t_n]\), and any measurable function \(g(x(t_1,w), \ldots, x(t_n,w))\), it is true that

\[\begin{split} \begin{aligned} \int\ & g\big(x(t_1,w), \ldots, x(t_n,w)\big)\, dP(w) = \\ \int\ & g\big(x(t_1 + \varepsilon, w), \ldots, x(t_n + \varepsilon, w)\big)\, dP(w), \ \text{ for every } \varepsilon \text{ in } \mathbb{R}. \end{aligned} \end{split}\]

Notice that strict stationarity implies covariance stationarity, but not vice versa. Throughout this book we work with the weaker, second-moment notion. The next chapter shows that the smoothness of the autocovariance function \(R(\tau)\) at the origin governs the continuity and differentiability of \(x(t)\) itself. 9. Characterizations of Mean Square Differentiability and Mean Square Continuity and 13. Locally Unpredictable Stochastic Processes return to that theme.