Scaling Exponent for Incremental Records

TL;DR

This paper studies the behavior of record increments in sequences of random variables, deriving an algebraic decay exponent for sequences with decreasing increments and analyzing distributions of records and increments.

Contribution

It introduces a novel analysis of decreasing increment sequences, deriving the decay exponent and characterizing the increment distribution with a power-law tail.

Findings

Fraction of sequences with decreasing increments decays as N^{-nu} with nu ≈ 0.318

Record distribution is narrow with an exponential tail

Increment distribution is broad with a power-law tail

Abstract

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.