Records from partial comparisons and discrete approximations

Abstract

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~\(f.\) 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~\(r \geq 1.\) 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.