Question

Let \(i=\sqrt{-1} .\) Define the integral of a complex-valued function with respect to Gaussian random measure as the sum of the integrals of the real and imaginary parts. Similarly, define the integral of a function with respeet to a complex random measure $$ \zeta(\mathbf{I})=\xi(\mathrm{I})+i \eta(\mathrm{I}), \quad \mathrm{I}=(s, t], \quad s

Short answer

The expectation \(E\left[\zeta\left(I_{1}\right) \overline{\left.\zeta\left(I_{2}\right)\right]}\right.\), where \(I_{i}=\left(s_{i}, t_{i}\right]\), can be computed with the given properties of \(\xi\) and \(\eta\) as: \[ E\left[\zeta\left(I_{1}\right) \overline{\left.\zeta\left(I_{2}\right)\right]}\right. = E\left[\xi(I_1) \xi(I_2)\right] + iE\left[\xi(I_1) \eta(I_2)\right] - iE\left[\eta(I_1) \xi(I_2)\right] + E\left[\eta(I_1) \eta(I_2)\right] \]

Step-by-step solution

Substitute the given definitions for \(\xi\) and \(\eta\)

Let's substitute the given definitions for \(\xi\) and \(\eta\) into the expression \( X_{n}=\int_{0}^{n} \cos n \omega Z^{(1)}(d \omega)+\int_{0}^{\pi} \sin n \omega Z^{(2)}(d \omega) \): \[ X_{n} = \int_{0}^{\pi} \cos{n\omega} \left(2\xi(s)\right) d\omega + \int_{0}^{\pi} \sin{n\omega} \left(\pm 2\eta(s)\right) d\omega \] Now, we have the expression for \(X_{n}\) in terms of \(\xi\) and \(\eta\). Moving on to the next step.

Obtain the representation \( X_{n}=\int_{-n}^{\pi} e^{-i_{n} \omega} \zeta(d \omega) \)

Now, using the definition of the integral of a complex-valued function with respect to the Gaussian random measure, we can express the integral as follows: \[ X_n = \int_{-n}^\pi \left( \cos{n\omega} \left(2\xi(s)\right) + i \sin{n\omega} \left(\pm 2\eta(s)\right) \right) d\omega \] Next, we'll express the integrand as a complex exponential using Euler's formula: \[ \begin{aligned} e^{-i_n \omega}(2\xi(s)\pm i 2\eta(s))&= (\cos{n\omega}-i\sin{n\omega})(2\xi(s) + \pm i2\eta(s)) \\ &= 2\xi(s)\cos{n\omega} \pm i2\eta(s)\sin{n\omega} \end{aligned} \] Thus, the representation of \(X_n\) can be written as: \[ X_n = \int_{-n}^\pi e^{-i_n \omega} \zeta(d\omega) \]

Compute the expectation\(E\left[\zeta\left(I_{1}\right) \overline{\left.\zeta\left(I_{2}\right)\right]}\right.\)

Now, let's compute the expectation \(E\left[\zeta\left(I_{1}\right) \overline{\left.\zeta\left(I_{2}\right)\right]}\right.\), where \(I_{i}=\left(s_{i}, t_{i}\right]\). Using the given properties of \(\xi\) and \(\eta\), we have: \[ E\left[\zeta\left(I_{1}\right) \overline{\left.\zeta\left(I_{2}\right)\right]}\right. = E\left[\left(\xi(I_1)+i\eta(I_1)\right) (\xi(I_2)-i\eta(I_2))\right] \] To compute this expectation, we'll use the linearity property of expectation: \[ E\left[\zeta\left(I_{1}\right) \overline{\left.\zeta\left(I_{2}\right)\right]}\right. = E\left[\xi(I_1) \xi(I_2) - i\eta(I_1) \xi(I_2) + i\xi(I_1) \eta(I_2) + \eta(I_1) \eta(I_2)\right] \] Applying the linearity property of expectation again, we get: \[ E\left[\zeta\left(I_{1}\right) \overline{\left.\zeta\left(I_{2}\right)\right]}\right. = \underbrace{E\left[\xi(I_1) \xi(I_2)\right]}_{\text{Term 1}} - \underbrace{iE\left[\eta(I_1) \xi(I_2)\right]}_{\text{Term 2}} + \underbrace{iE\left[\xi(I_1) \eta(I_2)\right]}_{\text{Term 3}} + \underbrace{E\left[\eta(I_1) \eta(I_2)\right]}_{\text{Term 4}} \] Given the observation \( \xi(I)=\xi(-I)\) and \(\eta(I)=-\eta(-I)\), we have: 1. \(E\left[\xi(I_1)\xi(I_2)\right]\) = \(E\left[\xi(-I_1)\xi(-I_2)\right]\) 2. \(E\left[\eta(I_1)\xi(I_2)\right]\) = \(-E\left[\eta(-I_1)\xi(-I_2)\right]\) 3. \(E\left[\xi(I_1)\eta(I_2)\right]\) = \(-E\left[\xi(-I_1)\eta(-I_2)\right]\) 4. \(E\left[\eta(I_1)\eta(I_2)\right]\) = \(E\left[\eta(-I_1)\eta(-I_2)\right]\) Computing each term, we obtain the final result for the expectation: \[ E\left[\zeta\left(I_{1}\right) \overline{\left.\zeta\left(I_{2}\right)\right]}\right. = E\left[\xi(I_1) \xi(I_2)\right] + iE\left[\xi(I_1) \eta(I_2)\right] - iE\left[\eta(I_1) \xi(I_2)\right] + E\left[\eta(I_1) \eta(I_2)\right] \]

Key concepts

Complex-Valued Function Integral

When dealing with stochastic processes, it's not uncommon to come across the integration of complex-valued functions, particularly in relation to random measures. The integral of a complex-valued function with respect to a Gaussian random measure elegantly illustrates how we can extend the concept of integration from the real to the complex domain. By defining this integral as the sum of the integrals of its real and imaginary parts, we retain the essence of integrating functions while adding the richness of complex analysis.

In practical terms, if we have a complex function expressed as \( f(s) = u(s) + iv(s) \), where \( u \) and \( v \) are real-valued functions, and we want to integrate this function with respect to a Gaussian random measure, we simply integrate \( u \) and \( v \) separately. This approach mirrors the process used in complex calculus and ensures that the stochastic integral respects the properties typical of complex functions. The integration results in a complex number whose real and imaginary parts are given by the integrals of \( u \) and \( v \), respectively. This process is essential for solving problems involving complex stochastic systems, such as the computation of Fourier transforms in noise processes.

For example, an integral like \( X_{n} = \int_{-n}^{\pi} e^{-in \omega} \zeta(d \omega) \) is calculated by taking the integral of the real part \( \cos{n\omega} \) and the imaginary part \( \sin{n\omega} \) of the exponential function separately, and then combining them to form the complex result.

Gaussian Random Measure

A Gaussian random measure is a fundamental concept in stochastic processes that is associated with Gaussian probability distributions. Gaussian measures are instrumental in building Gaussian processes, which are widely used for modeling various phenomena in fields such as finance, physics, and engineering.

Essentially, a Gaussian random measure assigns a normally distributed random variable to each measurable set in a given space, and these assignments adhere to the property of additivity. The key feature of a Gaussian random measure is that the integral with respect to it results in Gaussian random variables. For instance, if \( Z^{(1)}(d \omega) \) and \( Z^{(2)}(d \omega) \) in our previous example are considered Gaussian random measures, then their integrals with respect to \( \cos{n\omega} \) and \( \sin{n\omega} \) will yield Gaussian random variables which are components of the function \( X_{n} \).

In practice, when we decompose a Gaussian process into its sine and cosine components, we are often doing so using the framework provided by the Gaussian random measure. This decomposition allows us to not only study the properties of the process in a more granular way but also to perform operations such as Fourier transforms, which are crucial in signal processing and other analytical techniques.

Complex Random Measure

Building on the concepts of complex integrals and Gaussian random measures, we come to the concept of a complex random measure. A complex random measure is an advanced concept where both the real and imaginary components of the measure are random measures themselves, typically with some dependency structure.

In our exercise, the complex random measure \( \zeta(\mathbf{I}) \) is defined as \( \zeta(\mathbf{I})=\xi(\mathrm{I})+i \eta(\mathrm{I}), \quad \mathrm{I}=(s, t], \quad s
The significance of this concept is highlighted when we construct complex-valued stochastic processes, which are used to model systems where amplitude and phase variation are equally important, such as in the case of electromagnetic fields or quantum mechanics. The intricate relationship between \( \xi \) and \( \eta \) in the form of their respective symmetries that \( \xi(I)=\xi(-I) \) and \( \eta(I)=-\eta(-I) \) help in simplifying the computations. By considering both measures together, we gain a powerful tool for working with complex signals and can fully leverage the complexity in the data we are analyzing.