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

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.

21 pages, 7 figures

References in corpus (5)

• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• Universal Record Statistics of Random Walks and Lévy Flights
• Records in a changing world
• Integer Partitions and Exclusion Statistics
• Exact solution of a model of time-dependent evolutionary dynamics in a rugged fitness landscape

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