Records and sequences of records from random variables with a linear trend

TL;DR

This paper analyzes the probability of records in linear trend time series, focusing on asymptotic behaviors for different drift velocities, with applications in climatology and evolutionary biology.

Contribution

It provides new insights into the asymptotic behavior of record probabilities in linear trend series, extending understanding in climatology and biology contexts.

Findings

Asymptotic behavior characterized for small and large drift velocities.

Derived formulas for record occurrence probabilities in linear trend series.

Applications demonstrated in climatology and evolutionary biology.

Abstract

We consider records and sequences of records drawn from discrete time series of the form $X_{n}=Y_{n}+cn$, where the $Y_{n}$ are independent and identically distributed random variables and $c$ is a constant drift. For very small and very large drift velocities, we investigate the asymptotic behavior of the probability $p_n(c)$ of a record occurring in the $n$th step and the probability $P_N(c)$ that all $N$ entries are records, i.e. that $X_1 < X_2 < ... < X_N$. Our work is motivated by the analysis of temperature time series in climatology, and by the study of mutational pathways in evolutionary biology.