Convolution Bounds on Quantile Aggregation

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.

Funding: This work was supported by the Guangdong Provincial Key Laboratory of Mathematical Foundations for Artificial Intelligence [Grant 2023B1212010001]; The Chinese University of Hong Kong (Shenzhen) research startup fund [Grant UDF01003336]; Natural Sciences and Engineering Research Council of Canada [Grants CRC-2022-00141 and RGPIN-2024-03728]; Shenzhen Science and Technology Program [Grant RCBS20231211090814028]; National Science Foundation [Grants 1915967, 2118199, 2229011, CAREER CMMI-1834710, and IIS-1849280]; Air Force Office of Scientific Research [Grant FA9550-20-1-0397]; and National Natural Science Foundation of China [Grant 12401624].

Supplemental Material: All supplemental materials, including the computer code and data that support the findings of this study, are available at https://doi.org/10.1287/opre.2021.0765.