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 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 nth 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.