Scaling Exponent for Incremental Records
P.W. Miller and E. Ben-Naim
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 $I_N \sim N^{-\nu}$ as $N \rightarrow \infty$. We analyze the
case where the random variables are drawn from a uniform
distribution with compact support, and obtain the exponent $\nu =
0.317621\ldots$ 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.
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