Consider a strictly increasing and strictly concave utility function \(u\) and suppose that there exist \(N\) risky assets whose returns \(\left(\tilde{r}_{1}, \ldots, \tilde{r}_{N}\right)\) admit the representation

\[\tilde{r}_{n}=\sum_{\substack{k=1 \\ k eq n}}^{N} \lambda_{k} \tilde{r}_{k}+\tilde{\epsilon}_{n}, \quad \text { for all } n=1, \ldots, N\]

where \(\sum_{k=1, k eq n}^{N} \lambda_{k}=1\), for all \(n=1, \ldots, N\), and \(\tilde{\epsilon}_{n}\) is a random variable satisfying

\[\begin{equation*}\mathbb{E}\left[\tilde{\epsilon}_{n} \mid \tilde{r}_{1}, \ldots, \tilde{r}_{n-1}, \tilde{r}_{n+1}, \ldots, \tilde{r}_{N}\right]=\mathbb{E}\left[\tilde{\epsilon}_{n}\right] \tag{3.55}\end{equation*}\]

for all \(n=1, \ldots, N\). Show that \(w_{n}^{*}>0\) if and only if \(\mathbb{E}\left[\tilde{\epsilon}_{n}\right]>0\), for all \(n=1, \ldots, N\).