  We investigate records in a growing sequence of identical and independently
distributed random variables. The record equals the largest value in the
sequence, and our focus is on the increment, defined as the difference between
two successive records. We investigate sequences in which all increments
decrease monotonically, and find that the fraction I_N of sequences that
exhibit this property decays algebraically with sequence length N, namely I_N ~
N^{-nu} as N --> infinity. We analyze the case where the random variables are
drawn from a uniform distribution with compact support, and obtain the exponent
nu = 0.317621... using analytic methods. We also study the record distribution
and the increment distribution. Whereas the former is a narrow distribution
with an exponential tail, the latter is broad and has a power-law tail
characterized by the exponent nu. Empirical analysis of records in the sequence
of waiting times between successive earthquakes is consistent with the
theoretical results.
