  We consider a class of chance-constrained programs in which profit needs to
be maximized while enforcing that a given adverse event remains rare. Using
techniques from large deviations and extreme-value theory, we show how the
optimal value scales as the prescribed bound on the violation probability
becomes small and how convex programs emerge in the limit. We also use our
results to show that the popular CVaR approximation is asymptotically optimal
under light-tail assumptions, but it is sub-optimal in a heavy-tailed setting.
Then, we apply point process techniques and random set theory to study the
suboptimality gap in the Monte Carlo based scenario approach.
