Records in a changing world

TL;DR

This paper extends the theory of records to sequences of independent random variables with evolving distributions, revealing new asymptotic behaviors of record counts depending on distribution classes and showing correlations in record events.

Contribution

It introduces new results for record statistics in non-i.i.d. sequences with time-dependent distributions, broadening classical record theory.

Findings

Mean number of records varies with distribution class: logarithmic, squared logarithmic, or power-law growth.

Record events are correlated, reducing variance compared to the mean.

Simulations support theoretical predictions for different distribution classes.

Abstract

In the context of this paper, a record is an entry in a sequence of random variables (RV's) that is larger or smaller than all previous entries. After a brief review of the classic theory of records, which is largely restricted to sequences of independent and identically distributed (i.i.d.) RV's, new results for sequences of independent RV's with distributions that broaden or sharpen with time are presented. In particular, we show that when the width of the distribution grows as a power law in time $n$, the mean number of records is asymptotically of order $\ln n$ for distributions with a power law tail (the \textit{Fr\'echet class} of extremal value statistics), of order $(\ln n)^2$ for distributions of exponential type (\textit{Gumbel class}), and of order $n^{1/(\nu+1)}$ for distributions of bounded support (\textit{Weibull class}), where the exponent $\nu$ describes the behaviour of the distribution at the upper (or lower) boundary. Simulations are presented which indicate that, in contrast to the i.i.d. case, the sequence of record breaking events is correlated in such a way that the variance of the number of records is asymptotically smaller than the mean.