Rounding Effects in Record Statistics

TL;DR

This paper investigates how rounding in discretized continuous time series affects record-breaking events, revealing that rounding reduces new records and alters their statistical properties, especially for distributions with finite or thin upper tails.

Contribution

It provides a theoretical analysis of the impact of rounding on record statistics in discretized data, highlighting differences from continuous cases and long-term regularities.

Findings

Rounding reduces the number of new records in discretized series.

For finite upper limit distributions, the number of records is finite.

Distributions with thin upper tails lead to highly regular record sequences.

Abstract

We analyze record-breaking events in time series of continuous random variables that are subsequently discretized by rounding down to integer multiples of a discretization scale $\Delta>0$. Rounding leads to ties of an existing record, thereby reducing the number of new records. For an infinite number of random variables that are drawn from distributions with a finite upper limit, the number of discrete records is finite, while for distributions with a thinner than exponential upper tail, fewer discrete records arise compared to continuous variables. In the latter case the record sequence becomes highly regular at long times.