Limit laws for \(K\)-record times
Let \(X_ 1, X_ 2, \dots, X_ n\) be independent and identically distributed random variables. Let \(X_{r,n}\) be the \(r\)-th order statistic of the \(X_ j\), \(1\leq j\leq n\). Assuming that the variables \(X_ j\) come from an infinite sequence, let the rank \(R_ n\) of \(X_ n\) be defined by \(X_ n= X_{R_ n, n}\). For a given sequence \(k_ n\), \(n\geq 1\), of positive integers, \(K= \{k_ n\}\), record times \(L(n)\), \(n\geq 0\), are defined by the sequential procedure \(L(0)=0\) and \(L(n)= \inf \{m\): \(m>L(n-1)\), \(R_ m\geq m-k_ m+1\}\) for \(n\geq 1\). The authors introduce indicator variables associated with the sequence \(L(n)\) which are shown to be independent. This enables the authors to establish strong laws of large numbers, functional laws of the iterated logarithm and strong invariance principles for \(L(n)\). These give far reaching extensions of the by now classical results for record times which correspond to the case \(k_ n=1\) for all \(n\geq 1\). Besides the reduction to independent variables, the authors utilize the martingale properties of several transforms of the sequence \(L(n)\) for evaluating some moments.
functional laws of the iterated logarithm
scientific article; zbMATH DE number 605292
scientific article