We propose a general methodology for performing statistical inference within
a `rare-events regime' that was recently suggested by Wagner, Viswanath and
Kulkarni. Our approach allows one to easily establish consistent estimators for
a very large class of canonical estimation problems, in a large alphabet
setting. These include the problems studied in the original paper, such as
entropy and probability estimation, in addition to many other interesting ones.
We particularly illustrate this approach by consistently estimating the size of
the alphabet and the range of the probabilities.