  Extreme events can come either from point processes, when the size or energy
of the events is above a certain threshold, or from time series, when the
intensity of a signal surpasses a threshold value. We are particularly
concerned by the time between these extreme events, called respectively waiting
time and quiet time. If the thresholds are high enough it is possible to
justify the existence of scaling laws for the probability distribution of the
times as a function of the threshold value, although the scaling functions are
different in each case. For point processes, in addition to the trivial Poisson
process, one can obtain double-power-law distributions with no finite mean
value. This is justified in the context of renormalization-group
transformations, where such distributions arise as limiting distributions after
iterations of the transformation. Clear connections with the generalized
central limit theorem are established from here. The non-existence of finite
moments leads to a semi-parametric scaling law in terms of the sample mean
waiting time, in which the (usually unkown) scale parameter is eliminated but
not the exponents. In the case of time series, scaling can arise by considering
random-walk-like signals with absorbing boundaries, resulting in distributions
with a power-law "bulk" and a faster decay for long times. For large thresholds
the moments of the quiet-time distribution show a power-law dependence with the
scale parameter, and isolation of the latter and of the exponents leads to a
non-parametric scaling law in terms only of the moments of the distribution.
Conclusions about the projections of changes in the occurrence of natural
hazards lead to the necessity of distinguishing the behavior of the mean of the
distribution with the behavior of the extreme events.
