Records in a partially ordered set
Let S be a set equipped with a strict partial order \(``<''\) and let \(X_ 1,X_ 2,..\). be a sequence of independent, identically distributed S- valued random elements; \(X_ n\) is said to be a record iff \(X_ k<X_ n\), \(k=1,2,...,n-1.\) The authors define a hazard measure H on (S,\(s)\), where \(s\) is a \(\sigma\)-algebra of subsets of S. H(A), \(A\in s\), is the expected number of records that fall into A. The number \(N_ A\) of records in A is finite a.s. or infinite a.s. according as H(A) is finite or infinite. In the former case the authors calculate all the moments of \(N_ A\) and the probability law of \(N_ A\). They also show that the sequence of record values in A is a Markov chain and that for several sets A,B,... that are totally ordered the restrictions of the records process to the individual sets form independent processes. In the particular case of \(R^ 2\) the authors apply the general theory to the case of probability laws in the domain of attraction of a bivariate extreme-value law.
scientific article; zbMATH DE number 4111664