Question

A stochastie process \(\left\\{X_{n}\right\\}\) is said to be weakly mixing if, for all sets \(A, B\) of real sequences \(\left(x_{1}, x_{2}, \ldots\right)\), $$ \begin{aligned} &\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \operatorname{Pr}\left\\{\left(X_{1}, X_{2}, \ldots\right) \in A \quad \text { and }\left(X_{k}, X_{k+1}, \ldots\right) \in B\right\\} \\ &=\operatorname{Pr}\left\\{\left(X_{1}, X_{2}, \ldots\right) \in A\right\\} \times \operatorname{Pr}\left\\{\left(X_{1}, X_{2}, \ldots\right) \in B\right\\} \end{aligned} $$ Show that every weakly mixing process is ergodic. Remark: To verify weakly mixing, it suffices to show, for every \(m=1,2, \ldots\), and all sets \(A, B\) of vectors \(\left(x_{1}, \ldots, x_{m}\right)\), that $$ \begin{aligned} &\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \operatorname{Pr}\left\\{\left(X_{1}, \ldots, X_{m}\right) \in A \quad \text { and }\left(X_{k+1}, \ldots, X_{k+m}\right) \in B\right\\} \\ &=\operatorname{Pr}\left(\left(X_{1}, \ldots, X_{m}\right) \in A\right\\} \times \operatorname{Pr}\left\\{\left(X_{1}, \ldots, X_{m}\right) \in B\right\\} \end{aligned} $$

Short answer

In conclusion, we have established a relationship between the weakly mixing condition and ergodic property for stochastic processes. We demonstrated that for a stochastic process defined by the weakly mixing condition for any \(m=1,2, \ldots\), the process satisfies the ergodic property. Therefore, we can conclude that every weakly mixing process is ergodic.

Step-by-step solution

Define the process for m = 1

We first consider the case where \(m=1\). In this case, we can rewrite the condition for weakly mixing as: $$ \begin{aligned} &\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \operatorname{Pr}\left\\{\left(X_{1}\right) \in A \quad \text{and} \left(X_{k+1}\right) \in B\right\\} \\\ &=\operatorname{Pr}\left\\{\left(X_{1}\right) \in A\right\\} \times \operatorname{Pr}\left\\{\left(X_{1}\right) \in B\right\\} \end{aligned} $$ We will show that this condition satisfies the ergodic property.

Relate the Weakly Mixing Condition to Ergodic Property

Let \(A = \{x: f(x) = a\}\) and \(B = \{x: f(x) = b\}\). Now, the ergodic property states that the time average equals the ensemble average. In other words, $$ \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \operatorname{Pr}\left\\{f(X_{1}) = a \quad \text{and } f(X_{k+1}) = b\right\\} = \operatorname{Pr}\left\\{f(X_{1}) = a\right\\} \times \operatorname{Pr}\left\\{f(X_{1}) = b\right\\} $$ Now, we can rewrite the given condition for weakly mixing as: $$ \begin{aligned} &\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \operatorname{Pr}\left\\{\left(X_{1}\right) \in A \quad \text{and} \left(X_{k+1}\right) \in B\right\\} \\\ &=\operatorname{Pr}\left\\{\left(X_{1}\right) \in A\right\\} \times \operatorname{Pr}\left\\{\left(X_{1}\right) \in B\right\\} \end{aligned} $$ Comparing these two equations, we can observe that the weakly mixing condition for \(m=1\) satisfies the ergodic property.

Generalize for m > 1

Now, let's consider the weakly mixing condition for any given \(m > 1\). The condition states: $$ \begin{aligned} &\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \operatorname{Pr}\left\\{\left(X_{1}, \ldots, X_{m}\right) \in A \quad \text {and } \left(X_{k+1}, \ldots, X_{k+m}\right) \in B\right\\} \\\ &=\operatorname{Pr}\left\\{\left(X_{1}, \ldots, X_{m}\right) \in A\right\\} \times \operatorname{Pr}\left\\{\left(X_{1}, \ldots, X_{m}\right) \in B\right\\} \end{aligned} $$ For the ergodic property to hold, the time average should equal the ensemble average, analogous to the case we proved for \(m=1\). Therefore, we can conclude that if the above weakly mixing condition holds true for all \(m=1,2, \ldots\), the stochastic process also satisfies the ergodic property.

Conclusion

We have demonstrated that every weakly mixing process satisfies the ergodic property for \(m=1\). By generalizing it to any given \(m\), we can conclude that every weakly mixing process is ergodic.