1. **Problem Statement:** We want to verify if the expression for the fourth moment of the Wiener process \(\mathbb{E}[W_{t_1}W_{t_2}W_{t_3}W_{t_4}] = t_2 t_4 + 2 t_3 t_4\) makes sense under the condition \(t_1 > t_2 > t_3 > t_4 > 0\). 2. **Recall the properties of Wiener process:** - \(W_t\) is a Gaussian process with mean zero. - Covariance is given by \(\mathbb{E}[W_s W_t] = \min(s,t)\). 3. **Formula for the fourth moment of Gaussian variables:** For zero-mean Gaussian variables, the fourth moment can be expressed as sums of products of covariances: $$\mathbb{E}[W_{t_1}W_{t_2}W_{t_3}W_{t_4}] = \mathbb{E}[W_{t_1}W_{t_2}]\mathbb{E}[W_{t_3}W_{t_4}] + \mathbb{E}[W_{t_1}W_{t_3}]\mathbb{E}[W_{t_2}W_{t_4}] + \mathbb{E}[W_{t_1}W_{t_4}]\mathbb{E}[W_{t_2}W_{t_3}]$$ 4. **Evaluate each covariance term using \(\min(s,t)\):** - \(\mathbb{E}[W_{t_1}W_{t_2}] = \min(t_1,t_2) = t_2\) since \(t_1 > t_2\) - \(\mathbb{E}[W_{t_3}W_{t_4}] = \min(t_3,t_4) = t_4\) since \(t_3 > t_4\) - \(\mathbb{E}[W_{t_1}W_{t_3}] = t_3\) - \(\mathbb{E}[W_{t_2}W_{t_4}] = t_4\) - \(\mathbb{E}[W_{t_1}W_{t_4}] = t_4\) - \(\mathbb{E}[W_{t_2}W_{t_3}] = t_3\) 5. **Substitute these into the formula:** $$\mathbb{E}[W_{t_1}W_{t_2}W_{t_3}W_{t_4}] = t_2 t_4 + t_3 t_4 + t_4 t_3 = t_2 t_4 + 2 t_3 t_4$$ 6. **Interpretation:** The expression is consistent with the properties of the Wiener process and the Gaussian moment factorization. 7. **Conclusion:** Yes, the expression \(\mathbb{E}[W_{t_1}W_{t_2}W_{t_3}W_{t_4}] = t_2 t_4 + 2 t_3 t_4\) makes sense for \(t_1 > t_2 > t_3 > t_4 > 0\) because it correctly applies the covariance structure and Gaussian moment factorization of the Wiener process.