  Extreme events are by nature rare and difficult to predict, yet are often
much more important than frequent, typical events. An interesting counterpoint
to the prediction of such events is their retrodiction -- given a process in an
outlier state, how did the events leading up to this endpoint unfold? In
particular, was there only a single, massive event, or was the history a
composite of multiple, smaller but still significant events? To investigate
this problem we take heavy-tailed stochastic processes (specifically,the
symmetric, $\alpha$-stable L\'evy processes) as prototypical random walks. A
natural and useful characteristic scale arises from the analysis of processes
conditioned to arrive in a particular final state (L\'evy bridges). For final
displacements longer than this scale, the scenario of a single, long jump is
most likely, even though it corresponds to a rare, extreme event. On the other
hand, for small final displacements, histories involving extreme events tend to
be suppressed. To further illustrate the utility of this analysis, we show how
it provides an intuitive framework for understanding three problems related to
boundary crossings of heavy-tailed processes. These examples illustrate how
intuition fails to carry over from diffusive processes, even very close to the
Gaussian limit. One example yields a computationally and conceptually useful
representation of L\'evy bridges that illustrates how conditioning impacts the
extreme-event content of a random walk. The other examples involve the
conditioned boundary-crossing problem and the ordinary first-escape problem; we
discuss the observability of the latter example in experiments with
laser-cooled atoms.
