An example on the risk dependence and additivity of intensities in the theory of competing risks

Let vk(x) be the force of mortality when only risk Rk is acting in a population, and let muk(x) be the force of mortality due to Rk when all of the risks of death are acting, k = 1, ---, c. On the assumption of the independence of the risks of death, the sum of intensitives vk(x) equals the total force of mortality. This example shows that the equality can also hold when the risks of death are dependent and the forces of mortality, muk(x) and vk(x), differ.