Question

Show that no covariance stationary process \(\left\\{X_{n}\right\\}\) can satisfy the stochastic, difference equation \(X_{n}=X_{n-1}+\varepsilon_{n}\), when \(\left\\{\varepsilon_{n}\right\\}\) is a sequence of zero-mean uncorrelated random variables having a common positive variance \(\sigma^{2}>0\).

Short answer

The given stochastic difference equation \(X_{n}=X_{n-1}+\varepsilon_{n}\) contradicts the basic property of a covariance stationary process. As a result, no covariance stationary process can satisfy this equation.

Step-by-step solution

Understanding Covariance Stationary Process

Covariance stationary process is a type of stochastic process in which the mean and variance are constant over time, and the covariance between two time periods depends only on the distance or gap or lag between the two time periods and not the actual time at which the covariance is computed. In mathematical terms, a process \(\{X_{n}\}\) is covariance stationary if it satisfies the following conditions: 1) \(E[X_{n}]\) = \(\mu\) is independent of n. 2) \(Var[X_{n}]\) = \(\sigma^{2}\) is independent of n. 3) \(Cov[X_{n}, X_{n+k}]\) = \(\gamma_k\) is dependent only on k but not on n.

Analyzing the Given Stochastic Difference Equation

According to the problem, we have a stochastic difference equation as \(X_{n} = X_{n - 1} + \varepsilon_{n}\). If \(\varepsilon_{n}\) is a zero-mean uncorrelated and common variance \(\sigma^{2}>0\) sequence, then for \(n \geq 1\), one can find that \(X_{n}\) = \(X_{0}\) + \(\varepsilon_{1}\) + \(\varepsilon_{2}\) + ... + \(\varepsilon_{n}\). This sequence of random variables, \(\varepsilon_{n}\), results in \(X_{n}\) being a sum of these random variables, which makes it dependent on the number of variables (or the value of n).

Showing the contradiction

Now, the expected value of \(X_{n}\) is \(E[X_{n}]\) = \(E[X_{0}]\) + n * \(E[\varepsilon]\) = \(E[X_{0}]\) as \(E[\varepsilon]\) = 0. However, this contradicts the covariance stationary process property as the mean \(E[X_{n}]\) should be independent of n. But here, for different n, \(X_{n}\) turns out to be different. Therefore, no covariance stationary process can satisfy the given stochastic difference equation.