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Statistics of superior records.

E Ben-Naim¹, P L Krapivsky

¹Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 17, 2013

Summary

We analyzed random variable sequences, finding that "superior sequences" with consistently high records decay algebraically with length. The decay rate depends on the distribution

Area of Science:

Probability Theory
Statistical Analysis
Random Processes

Background:

Understanding the statistical properties of records in random sequences is crucial for analyzing time series data.
The concept of running records (maximum to date) is fundamental in various fields, including finance and geophysics.

Purpose of the Study:

To investigate the statistical properties of "superior sequences" where all running records exceed the expected average.
To determine the decay rate of superior and "inferior sequences" (records below average) as sequence length increases.
To explore the relationship between the parent distribution's tail and the decay exponent.

Main Methods:

Mathematical analysis of sequences of identical and independently distributed random variables.

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Derivation of algebraic decay laws for the fraction of superior and inferior sequences.

Calculation of the decay exponent (β for superior, α for inferior) as a root of an integral equation.

Main Results:

The fraction of superior sequences S(N) decays algebraically as S(N)~N^(-β) for large N.
The decay exponent β is determined by the tail of the parent distribution and is a non-trivial root of an integral equation.
For a uniform distribution, β was found to be approximately 0.450265.
The fraction of inferior sequences I(N) also decays algebraically, I(N)~N^(-α), with a different exponent α.

Conclusions:

The statistical measures developed provide insights into the long-term behavior of random sequences.
The tail of the parent distribution critically influences the decay exponents of superior and inferior sequences.
These statistical measures have potential applications in analyzing real-world data, such as earthquake occurrences.