Records from partial comparisons and discrete approximations

In this paper we study records obtained from partial comparisons within a sequence of independent and identically distributed (i.i.d.) random variables, indexed by positive integers, with a common density~$f\mathrm{.}$ Our main result is that if the comparison sets along a subsequence of the indices satisfy a certain compatibility property, then the corresponding record events are independent. Moreover, the record event probabilities do not depend on the density~$f$ and we obtain closed form expressions for the distribution of~$r^{th}$ record value for any integer~$r\ge 1.$ Our proof techniques extend to the discrete case as well and we estimate the difference in record event probabilities associated with a continuous random variable~$X$ and its discrete approximations.