On the differentiation of random measures with respect to homothecy invariant convex bases

For every homothecy invariant convex density differentiation basis $B$ in $\mathbb R^d$, we characterize sequences of weights $w=(w_j)_{j\in \mathbb N}$ for which the random measures $\mu_{w,\theta }=\sum_{j=1}^\infty w_j \delta _{\theta_j}$ are differentiable with respect to the basis $B$ for almost every selection of a sequence of points $\theta_1,\theta_2,\ldots $ from the unit cube $[0,1]^d$.