Abstract In this paper, we propose a method based on distance correlation theory to measure and test nonlinear dependence between a set-valued random variable and a random vector. This distance-based measure of dependence takes a value of zero if and only if the set-valued random variable and the random vector are independent. The proposed method thus effectively extends the scope of distance correlation from real-valued random vectors to set-valued random variables. This extension can have many potential applications in economics and finance. We then apply the proposed method to measure and test for the association between three salient cryptocurrency characteristics – namely, market capitalization, liquidity, and investor attention – and future returns and volatility, which are summarized by a random interval, in the cross section of over one thousand cryptocurrencies. We find that this association is very strong and it tends to be persistent over time in our sample.
Ba Chu (2026) studied this question.