Records from partial comparisons and discrete approximations

TL;DR

This paper investigates the independence and distribution of records from partial comparisons in i.i.d. sequences, providing density-independent formulas and extending results to discrete approximations.

Contribution

It establishes conditions for independence of record events from partial comparisons and derives explicit distributions, including for discrete approximations.

Findings

Record events are independent under certain compatibility conditions.

Record event probabilities are density-independent.

Explicit formulas for the distribution of the r-th record value.

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.