  Quantile aggregation with dependence uncertainty has a long history in
probability theory with wide applications in finance, risk management,
statistics, and operations research. Using a recent result on inf-convolution
of quantile-based risk measures, we establish new analytical bounds for
quantile aggregation which we call convolution bounds. Convolution bounds both
unify every analytical result available in quantile aggregation and enlighten
our understanding of these methods. These bounds are the best available in
general. Moreover, convolution bounds are easy to compute, and we show that
they are sharp in many relevant cases. They also allow for interpretability on
the extremal dependence structure. The results directly lead to bounds on the
distribution of the sum of random variables with arbitrary dependence. We
discuss relevant applications in risk management and economics.
