On the asymptotic shape of quantile surfaces

This article is concerned with the asymptotic shape of quantile surfaces, defined as the set of quantiles at a given level $\alpha$ generated by a controlled one-dimensional distribution. Specifically, when the distribution arises as a linear combination of log-normal random variables and the control is a vector of positive coefficients, we prove that quantile surfaces are globally concave in the left tail ($\alpha \to 0$) and globally convex in the right tail ($\alpha \to 1$). Moreover, these surfaces exhibit asymptotic separation of scale and shape.