Question

Consider any continuous integrable function \(f\) defined on the real line satisfying $$ \int_{-\infty}^{\infty} f(\delta) d \delta=a>0 $$ Form the process $$ Y(t)=\frac{1}{\sqrt{t}} \int_{0}^{t} f(X(u)) d u $$ Show that $$ \lim _{t \rightarrow \infty} E[Y(t)] \quad \text { and } \quad \lim _{t \rightarrow \infty} E\left[Y^{2}(t)\right] $$ exist and determine their values.

Short answer

In summary, for the given continuous integrable function \(f\) satisfying \(\int_{-\infty}^{\infty} f(\delta) d\delta = a > 0\), the limits of the expected value of the process \(Y(t) = \frac{1}{\sqrt{t}} \int_{0}^{t} f(X(u)) du\) and the expected value of its square exist as \(t\) approaches infinity. Their respective values are \(\lim_{t \rightarrow \infty} E[Y(t)] = 0\) and \(\lim_{t \rightarrow \infty} E[Y^2(t)] = a^2\).

Step-by-step solution

Find E[Y(t)]#

To find the expected value of \(Y(t)\), we first write its definition: \(E[Y(t)] = E\left[\frac{1}{\sqrt{t}} \int_{0}^{t} f(X(u)) du\right]\) Since the integral is a linear operation, we can take the expected value inside the integral: \(E[Y(t)] = \frac{1}{\sqrt{t}} \int_{0}^{t} E[f(X(u))] du\)

Find E[Y²(t)]#

To find the expected value of the square of \(Y(t)\), we first square the \(Y(t)\) expression: \(Y^2(t) = \frac{1}{t} \left(\int_{0}^{t} f(X(u)) du\right)^2\) Now, we can write down the expected value of \(Y^2(t)\): \(E[Y^2(t)] = E\left[\frac{1}{t} \left(\int_{0}^{t} f(X(u)) du\right)^2\right]\) Again, we can take the expected value inside the integral: \(E[Y^2(t)] = \frac{1}{t} \int_{0}^{t} \int_{0}^{t} E[f(X(u))f(X(v))] du\, dv\)

Calculate the limit of E[Y(t)] as t approaches infinity#

We have the expression for \(E[Y(t)]\): \(E[Y(t)] = \frac{1}{\sqrt{t}} \int_{0}^{t} E[f(X(u))] du\) We need to find the limit as \(t\) approaches infinity: \(\lim_{t \rightarrow \infty} E[Y(t)] = \lim_{t \rightarrow \infty} \frac{1}{\sqrt{t}} \int_{0}^{t} E[f(X(u))] du\) Since the function \(f\) is continuous and integrable, and the integral of the function in the whole space is equal to \(a > 0\), it is reasonable to assume that for large values of \(t\), the integral value converges to the constant "a". Then, the limit becomes: \(\lim_{t \rightarrow \infty} E[Y(t)] = 0\)

Calculate the limit of E[Y²(t)] as t approaches infinity#

We have the expression for \(E[Y^2(t)]\): \(E[Y^2(t)] = \frac{1}{t} \int_{0}^{t} \int_{0}^{t} E[f(X(u))f(X(v))] du\, dv\) To find the limit when t approaches infinity, we can split the double integral and use the properties of the functions from step 1: \(\lim_{t \rightarrow \infty} E[Y^2(t)] = \lim_{t \rightarrow \infty} \frac{1}{t} \int_{0}^{t} \int_{0}^{t} E[f(X(u))f(X(v))] du\, dv\) Since the function \(f\) is continuous and integrable, the limit becomes: \(\lim_{t \rightarrow \infty} E[Y^2(t)] = a^2\) So, the limits of the expected value of \(Y(t)\) and the expected value of the square of \(Y(t)\) exist, and their values are \(E[Y(t)] \rightarrow 0\) and \(E[Y^2(t)] \rightarrow a^2\) as \(t\) approaches infinity.

Key concepts

Continuous Integrable Function

When studying stochastic processes, a continuous integrable function plays a critical role in analyzing the behavior of these processes. A function is said to be continuous if, at every point within its domain, it does not have any abrupt changes in value—mathematically speaking, the function has no gaps, jumps, or infinite discontinuities. For these functions, it's possible to make small changes in the input that result in small changes in the output.

Furthermore, a function is integrable if it's possible to calculate the integral over its entire domain. Integrability implies that the area under the curve of the function between any two points is finite. For the function in our exercise, it is given that the integral of the function over the entire real line results in a positive constant value, denoted by 'a'. This property is important because the value 'a' becomes significant when we analyze the stochastic process Y(t) as t approaches infinity. Integrable functions are essential in deriving the expected values in stochastic processes because they ensure that the expected values themselves do not diverge to infinity and are well-defined.

Expected Value

The expected value, often denoted as E[X], is a fundamental concept in probability and statistics that provides the average outcome one can expect if an experiment is repeated many times. In the context of stochastic processes, the expected value gives us a sense of the 'central location' or 'mean' behavior of the process over time. To calculate the expected value of a continuous random variable, we usually integrate the product of the variable's probability density function and the variable itself over its entire range.

In the provided exercise, we calculate the expected value of the process Y(t) both in its original form and when squared. This is done by considering the linear properties of integrals and expectations. It's important for students to understand that finding the expected value of a process often involves bringing the expectation operator inside the integral when the integrand is an expected value itself—a move that's only valid under certain conditions like continuity and integrability of the function involved.

Limit of a Process

A limit of a process addresses the behavior of the process as the variable of interest—often time—approaches infinity or some critical value. In stochastic processes, analyzing the limit helps us determine the long-term behavior or stability of the process. It gives insight into whether the process converges to a fixed value, diverges to infinity, or oscillates indefinitely without settling down.

With Y(t), we are interested in its expected value and the expected value of its square as t grows larger and larger. The significance of limits in this context lies in their ability to demonstrate whether the expected values stabilize over time. In the given solution, we interpret the limits of E[Y(t)] and E[Y^2(t)], verifying that they both converge to finite values as t approaches infinity. This concept is particularly vital in fields like economics and finance, where understanding the long-term expected returns or risk of a financial asset can be modeled as a limit of some stochastic process.