Observations occurring between successive record times and within a distance a > 0 of the current record value are called near-records. Limit theorems for the number ξ_n (a) of near records are found for cases in which the parent distribution lies in a maximal domain of attraction and α is a function of n. Corollaries are indicated for numbers of near-k-records and sums of near-records. If the parent law is thin-tailed and a is constant, then a centered and normed version of log ξ_n (a) has a limit law under appropriate conditions.
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