Question

Let \(\left\\{\xi_{n}\right\\}\) be a zero-mean covariance stationary process having covariance function $$ E\left[\xi_{n} \xi_{m}\right]= \begin{cases}1, & n=m \\ \rho, & n \neq m\end{cases} $$ where \(0<\rho<1\). Show that \(\left\\{\xi_{n}\right\\}\) has the representation \(\xi_{n}=U+\eta_{n}\), where \(U, \eta_{1}, \eta_{2}, \ldots\) are zero-mean, uneorrelated random variables, \(E\left[U^{2}\right]=\rho\), and \(E\left[\eta_{k}^{2}\right]=1-\rho\) Hint: Use the mean square ergodic theorem to define \(U=\lim \left(\xi_{t}+\cdots+\xi_{n}\right) / n\). Set \(\eta_{n}=\xi_{n}-U\) and compute \(E\left[U \xi_{n}\right], E\left[U^{2}\right]\), and \(E\left[\eta_{n} \eta_{m}\right]\)

Short answer

We have shown that the given stationary process \(\xi_n\) has the representation \(\xi_n = U + \eta_n\), where \(U\), \(\eta_1\), \(\eta_2,\ldots\) are zero-mean, uncorrelated random variables, \(E[U^2] = \rho\), and \(E[\eta_k^2] = 1-\rho\). This was achieved by defining \(U\) using the Mean Square Ergodic Theorem, setting \(\eta_n = \xi_n - U\), and computing the expectations \(E[U\xi_n]\), \(E[U^2]\), and \(E[\eta_n\eta_m]\) to verify that they satisfy the conditions in the problem statement.

Step-by-step solution

Define U with Mean Square Ergodic Theorem

To begin, define \(U\) the following way, using the Mean Square Ergodic Theorem: \[U = \lim_{n \to \infty}\frac{\xi_1 + \xi_2 + \cdots + \xi_n}{n}\]

Define ηn

Next, define \(\eta_n\) as: \[\eta_n = \xi_n - U\]

Calculate E[Uξn]

Now, we need to compute the expectation \(E[U\xi_n]\) which is given by: \begin{align*} E[U\xi_n] &= E\left[\left(\lim_{n \to \infty}\frac{\xi_1 + \xi_2 + \cdots + \xi_n}{n}\right) \xi_n\right] \end{align*} Since the process \(\{\xi_n\}\) is mean square ergodic, we can interchange the expectation and limit: \[E[U\xi_n] = \lim_{n \to \infty}\frac{E[\xi_1\xi_n] + E[\xi_2\xi_n] + \cdots + E[\xi_n\xi_n]}{n}\] According to the covariance function, we have: \[E[\xi_i\xi_n] = \begin{cases} 1, & i=n \\ \rho, & i \neq n \end{cases} \] So, we can rewrite the expectation: \[E[U\xi_n] = \lim_{n \to \infty}\frac{(n-1)\rho + 1}{n}\] As \(n \to \infty\), the expectation converges to \(\rho\).

Calculate E[U^2]

Next, calculate the expectation \(E[U^2]\), which is given by: \begin{align*} E[U^2] &= E\left[\left(\lim_{n \to \infty}\frac{\xi_1 + \xi_2 + \cdots + \xi_n}{n}\right)^2\right] \end{align*} Again, since the process \(\{\xi_n\}\) is mean square ergodic, we can interchange the expectation and limit: \[E[U^2] = \lim_{n \to \infty}\frac{E[\xi_1^2] + E[\xi_2^2] + \cdots + E[\xi_n^2] + 2\sum_{i < j} E[\xi_i\xi_j]}{n^2}\] Using the covariance function: \[E[U^2] =\lim_{n \to \infty}\frac{n + 2(n-1)\rho}{n^2}\] As \(n \to \infty\), the expectation converges to \(\rho\).

Calculate E[ηnηm]

Finally, calculate the expectation \(E[\eta_n\eta_m]\), which is given by: \begin{align*} E[\eta_n\eta_m] &= E[(\xi_n - U)(\xi_m - U)]\\ &= E[\xi_n\xi_m - U\xi_n - U\xi_m + U^2] \end{align*} Using the results from steps 3 and 4, and the given covariance function, we have: \begin{align*} E[\eta_n\eta_m] &= E[\xi_n\xi_m] - E[U\xi_n] - E[U\xi_m] + E[U^2] \end{align*} For \(n = m\), we get: \[E[\eta_n^2] = 1 - \rho - \rho + \rho = 1-\rho\] For \(n \neq m\), we get: \[E[\eta_n\eta_m] = \rho - \rho - \rho + \rho = 0\] With the calculated expectations satisfying the conditions in the problem statement, we have shown that the given stationary process has the representation \(\xi_n = U + \eta_n\) where \(U\), \(\eta_1\), \(\eta_2,\ldots\) are zero-mean, uncorrelated random variables, \(E[U^2] = \rho\), and \(E[\eta_k^2] = 1-\rho\).

Key concepts

Covariance Stationary Process

A covariance stationary process, also known as a weakly or second-order stationary process, is a stochastic process that satisfies specific statistical properties. For a process to be classified as covariance stationary, it must have a constant mean, constant variance, and a covariance between any two terms that depends solely on the time difference between them, not on the actual time at which the covariance is computed.
For instance, in the exercise provided, the process \(\{\xi_{n}\}\) is stated to be zero-mean covariance stationary, implying its mean is zero for all time periods. The covariance function given where \(E[\xi_{n} \xi_{m}] = 1\) for \(n=m\) and \(E[\xi_{n} \xi_{m}] = \rho\) for \(n eq m\), with \(0<\rho<1\), indicates that the covariance between any two different terms does not depend on their position in time but only on the fact that the terms are different.
Understanding the concept of a covariance stationary process is crucial in time series analysis as it enables the use of various statistical tools for forecasting and hypothesis testing. To improve grasp of this concept in exercises, one could consider visually plotting multiple realizations of the process and discussing real-world examples where such processes are encountered.

Mean Square Ergodic Theorem

The mean square ergodic theorem is a fundamental principle in the field of ergodic theory, which is a branch of statistics and probability theory. This theorem relates to the behavior of averages of random processes and states that, under certain conditions, the time average of a sequence of random variables will converge to their expected value, provided the average is taken over a long enough period.
The theorem is pivotal when dealing with stochastic processes where it provides a bridge between ensemble averages (averages taken over different realizations of a process) and time averages (averages taken over time for a single realization of the process). In simpler terms, it essentially says that in the long run, the average behavior of a system can be deduced by looking at one sufficiently long sample of the system's output.
To apply this theorem in the given exercise, the student is instructed to define \(U\) as the limit of the average of the sequence \(\xi_{n}\) as \(n\) goes to infinity. This approach can be enhanced by including illustrations on convergence and divergence of sequences in random processes and giving practical scenarios that showcase ergodic behavior.

Random Variables

Random variables are a foundational concept in probability theory and statistics. In essence, a random variable is a numerical descriptor of the outcomes of a random phenomenon. It's a variable whose possible values are numerical outcomes from a random process or experiment.
The exercise introduces \(U\), \(\eta_{1}\), \(\eta_{2}\), ..., as zero-mean uncorrelated random variables. The concept of 'zero-mean' signifies that the expected value (or long-term average, or probability-weighted average of all possible values) of these variables is zero. Being 'uncorrelated' indicates that these variables do not influence each other; the occurrence of one does not affect the probability distribution of the other.
For students to gain a deeper understanding, it is beneficial to discuss the implications of a variable being zero-mean and uncorrelated. Insights can be drawn from everyday decision-making scenarios where outcomes can be perceived as random variables – for example, discussing how flipping a fair coin can be modeled by a random variable with two outcomes. Simplifying the concept by detailing real-life analogies can help students relate to the statistical properties of random variables more intuitively.