Four deviations suffice for rank 1 matrices (Q2213767)

Consider any independent scalar random variables \(\xi_1, \ldots, \xi_n\) with finite support. Let \(u_1, \ldots, u_n \in\mathbb{R}^m\) and \(\sigma^2=|| \sum_{i=1}^n \mathrm{Var}[\xi_i](u_iu_i^*)^2||\). Then there exists a choice of outcomes \(\varepsilon_1,\ldots,\varepsilon_n\) in the support of \(\xi_1, \ldots, \xi_n\) such that \(||\sum_{i=1}^n \mathbb{E}[\xi_i]u_iu_i^*-\sum_{i=1}^n \varepsilon_iu_iu_i^*||\leq 4\sigma.\)