Abstract
We study d-records in sequences generated by independent geometric random variables and derive explicit and asymptotic formulæ for expectation and variance. Informally speaking, a d-record occurs, when one computes the d-largest values, and the variable maintaining it changes its value while the sequence is scanned from left to right. This is done for the "strict model," but a "weak model" is also briefly investigated. We also discuss the limit q → 1 (q the parameter of the geometric distribution), which leads to the model of random permutations.
Highlights
Records of a sequence of elements x1 . . . xn are a well studied subject (Nev01): The sequence is read from left to right, and whenever an element is encountered which is larger than the previously seen ones, we speak of a record
We study the instance of weak d–records, i.e., we replace X > Ci in the above algorithm by X ≥ Ci
Apart from the ungainly constant, that we no longer mention explicitly, we find that the contribution of log2 n cancels out, and the term logQ n + constant + δV (logQ n) has as a factor qd)22 γ L
Summary
Records (left–to–right maxima) of a sequence of elements x1 . . . xn are a well studied subject (Nev01): The sequence is read from left to right, and whenever an element is encountered which is larger than the previously seen ones, we speak of a record. We count how often the variable Cd changes its value This is what we will call number of d–records in this paper. We assume that the elements n ∈ N are drawn independently from a geometric distribution: P{X = k} = pqk−1, with p + q = 1 This is a situation for which we can still derive attractive results, and consider the limit q → 1 as well. The last value of the variable Cd for the computation of d–records is the d–largest value.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
