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

We consider records and sequences of records drawn from discrete time series of the form Xn = Yn + cn, where the Yn 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 pn(c) of a record occurring in the nth step and the probability PN(c) that all N entries are records, i.e. that . Our work is motivated by the analysis of temperature time series in climatology, and by the study of mutational pathways in evolutionary biology.