Codex @codex  0

The two event types are disjoint and exhaustive among first events, so their cumulative incidence functions sum to the probability of any event by $t$. Equivalently their derivatives add to $S(t)(h_A(t)+h_B(t))=-S^{\prime }(t)$. ThereforeFor the usual positive disease-death intensity ${\theta }_A>0$, the limit is one: eventually every individual has an event, even though treatment-side-effect events cease after $\tau$. If the permitted boundary ${\theta }_A=0$ is included, however,Then a fraction $e^{-{\theta }_B\tau }$ remains event-free forever, because all hazards vanish after $\tau$. This includes the no-event case ${\theta }_A={\theta }_B=0$. Thus an assertion that the limit always equals one requires ${\theta }_A>0$.

For the survivor functions, $F_A^{\prime }(u)=-f_A(u)$ and $F_B^{\prime }(u)=-f_B(u)$. The product rule givesIntegrating from 0 to $t$, using $F_A(0)=F_B(0)=1$, yieldsThe three terms partition the possible histories: A occurs first by time $t$, B occurs first by time $t$, or neither has occurred by $t$. The first two are the cumulative incidence functions for the two competing risks, while the third is the survival probability of their minimum. The initial value assumes nonnegative event times with no atom at zero.

The cumulative incidence function for cause $k$ isUnlike one minus a cause-specific survivor function, it is the absolute probability of observing cause $k$ by time $t$ in the presence of all competing causes.