Question

Let \(\\{X(t),-\infty

Short answer

In summary, to prove the properties of the weakly stationary process, we first showed that \(\widetilde{R}(w) = \widetilde{R}(-w)\) by using the evenness of the covariance function R(s). Then, we used the given formula relating R(s) and \(\widetilde{R}(w)\) to show that \(\int_{-\infty}^{\infty} \widetilde{R}(w) d w = 2 \pi E\left[X^{2}(t)\right]\).

Step-by-step solution

Prove that \(\widetilde{R}(w) = \widetilde{R}(-w)\)

To prove this property, we will use the fact that the covariance function R(s) is an even function, which is true for any weakly stationary process: $$ R(s) = \operatorname{Cov}(X(t), X(t + s)) = \operatorname{Cov} (X(t-s), X(t)) = R(-s) $$ Now, let's express \(\widetilde{R}(w)\) and \(\widetilde{R}(-w)\) using the given formula, and equate them: $$ \widetilde{R}(w) = \widetilde{R}(-w) $$ $$ \frac{1}{2 \pi} \int_{-\infty}^{\infty} R(s) e^{iws} ds = \frac{1}{2 \pi} \int_{-\infty}^{\infty} R(s) e^{-iws} ds $$ Since R(s) is an even function, and the integral is over the entire real line, we have: $$ \int_{-\infty}^{\infty} R(s) e^{iws} ds = \int_{-\infty}^{\infty} R(s) e^{-iws} ds $$ Thus, we have proved that \(\widetilde{R}(w) = \widetilde{R}(-w)\), as required.

Prove that \(\int_{-\infty}^{\infty} \widetilde{R}(w) d w = 2 \pi E\left[X^{2}(t)\right]\)

To prove this, we should first substitute R(s) in the given formula and evaluate the integral at s = 0: $$ R(0) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \widetilde{R}(w) e^{iw(0)} dw $$ $$ R(0) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \widetilde{R}(w) dw $$ We know that R(0) represents the variance of the process at any point, since the mean is constant and does not depend on time: $$ R(0) = E[X^2(t)] - (E[X(t)])^2 = E[X^2(t)] $$ Therefore, the equation becomes: $$ E[X^2(t)] = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \widetilde{R}(w) dw $$ Multiplying both sides by \(2\pi\), we obtain the desired result: $$ \int_{-\infty}^{\infty} \widetilde{R}(w) dw = 2 \pi E[X^2(t)] $$ In conclusion, we have shown that \(\widetilde{R}(w)=\widetilde{R}(-w)\), and also proved the given formula relating the power spectral density function \(\widetilde{R}(w)\) and the process's second moment (or the expectation of the square) \(E[X^2(t)]\).

Key concepts

Weakly Stationary Processes

In the realm of time series analysis and signal processing, the concept of weakly stationary processes is paramount. Such a process, also referred to as wide-sense stationary, possesses statistical properties that are invariant with respect to time shifts. This means that the mean and autocovariance of the process do not change over time.

More precisely, for a stochastic process to be weakly stationary, two conditions must be met: first, the expected value or mean of the process must be constant and independent of time, and second, the autocovariance function, which measures the dependence between two points of the process at different times, must depend only on the time difference and not on the actual times themselves.

This characteristic of weak stationarity is what allows us to employ powerful analytical tools, such as the Fourier transform, to study the underlying structure of these processes in the frequency domain through their power spectral density.

Covariance Function

Delving deeper into the second condition for weak stationarity, we encounter the covariance function, denoted by R(s). This function is a statistical tool used to quantify the relationship between two points in a stochastic process across time. It is defined as the covariance of the process at two different times, which in mathematical terms is \( \text{Cov}(X(t), X(t+s)) \).

Illustrating the Covariance Function

• It measures the extent to which deviations of a process at different times are related or the degree to which they 'move together'.
• The covariance function is central in understanding the temporal structure of a stochastic process.
• For weakly stationary processes, the covariance function simplifies significantly, since it only depends on the time lag s and not on the specific time t.

Understanding the covariance function is essential because it directly relates to the power spectral density, which acts as a bridge to the frequency domain analysis of the process.

Stochastic Process

At the core of our discussion lies the concept of a stochastic process. A stochastic process is a collection of random variables indexed by time, representing the evolution of some random phenomenon. It is essentially a mathematical model used to describe systems that evolve in a random, seemingly unpredictable manner over time.

A simple yet relatable example is the stock market, where the prices of stocks change continuously and involve chance. Here, the price of a stock at any given time can be considered as a random variable, and the sequence of prices over time forms a stochastic process.

Stochastic Process in Context

• Although the future of a stochastic process cannot be predicted precisely, the process might still adhere to a set of probabilistic rules, which provide structure to its behavior.
• Studying stochastic processes helps us make meaningful inferences about future behavior, understand potential patterns, and estimate probabilities of various outcomes.

As we dig into their analysis, terms like 'weakly stationary', 'covariance function', and 'power spectral density' are the tools we use to capture and quantify the nuances of these random yet analyzable models.

Combining these concepts, we achieve a comprehensive view of stochastic processes, which is integral for statistical modeling and forecasting in numerous disciplines, from finance to physics and beyond.