Near-record observations are the ones that occur between successive record times and within a fixed distance of the current record value. In this paper, the concept of near-record observations is generalized to the notion of observations that fall into a random region determined by a given record and a Borel set. Description of the distribution of the number of such observations is provided and asymptotic behavior of this number is investigated. In addition, some applications of the new results to derive exact and asymptotic properties of inter-record times and of numbers of repetitions of records are presented.
