= Solution
For mutually exclusive event types, let $T$ be the first event time and $J$ its type. The <cause-specific hazard> for cause $j$ is
$$
h_j(t)=\lim_{\Delta\downarrow0}\frac{P(t\le T<t+\Delta,J=j\mid T\ge t)}{\Delta}.
$$
It is a rate conditional on having had no event of any cause. The <cumulative incidence function>, also called the <cumulative risk function>, is the actual <probability> $F_j(t)=P(T\le t,J=j)$.
The total <hazard function> is $h(t)=\sum_kh_k(t)$, giving $S(t)=\exp\{-\int_0^t\sum_kh_k(u)\,du\}$. Surviving every cause to time $u$ and then experiencing cause $j$ gives
$$
\boxed{F_j(t)=\int_0^tS(u)h_j(u)\,du=\int_0^t\exp\left\{-\int_0^u\sum_kh_k(v)\,dv\right\}h_j(u)\,du.}
$$
In <competing risks>, $F_j$ is generally not $1-e^{-\int_0^th_j}$, because competing events remove individuals before they can experience cause $j$. This formula does not require assuming independent latent failure times for the different causes.
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