Precise asymptotics for record times and the associated counting process

Precise asymptotics have been proved for sums like ∑n=1∞nr/p−2P(|Sn|⩾εn1/p) as ε↘0, where {Sn, n⩾1} are partial sums i.i.d. random variables, and, more recently, for renewal counting processes and first passage time processes of random walks. The present paper is devoted to analogous results for the record times and the associated counting process of i.i.d. absolutely continuous random variables.