  This paper enhances our comprehension of the Distributional Synthetic Control
(DSC) proposed by Gunsilius (2023), focusing on its asymptotic properties. We
first establish the DSC estimator's asymptotic optimality. The essence of this
optimality lies in the treatment effect estimator given by DSC achieves the
lowest possible squared prediction error among all potential treatment effect
estimators that depend on an average of quantiles of control units. We also
establish the convergence of the DSC weights when some requirements are met, as
well as the convergence rate. A significant aspect of our research is that we
find DSC synthesis forms an optimal weighted average, particularly in
situations where it is impractical to perfectly fit the treated unit's
quantiles through the weighted average of the control units' quantiles. To
corroborate our theoretical insights, we provide empirical evidence derived
from simulations.
