2022/10/16 by Daniel Q. Naiman, Naiman, Daniel Q., Fred Torcaso +1
Mathematics · #60F05 (Primary) 60D05 (Secondary) #FOS: Mathematics #Fuzzy Systems and Optimization #Probability (math.PR) #Probability and Statistical Research #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2210.08426
openalex publication_date 2022/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Xi,i=0,1,… be a sequence of iid random variables whose distribution is continuous. Associated with this sequence is the sequence (i,Xi),i=0,1,…. Let \cal Rn denote the set of Pareto optimal elements of \ (i,Xi):i=0,…,n\. We refer to the elements of \cal Rn as the current records at time n, and we define Rn=\vert \cal Rn\vert, the number of such records. Observe that Rn has \1,…,n+1\ as its support. When (n,Xn) is realized, it is a Pareto optimal element of \ (i,Xi)~:~i=0,…,n\ and \cal Rn \backslash (n,Xn) ⊂ \cal Rn-1. Then we refer to those elements of \cal Bn = \cal Rn-1 \backslash \cal Rn as the records broken at time n. Let Bn= \vert \cal Bn \vert. We show that P[Bn = k] → 1/2k+1 for k=0,1,2,….