The asymptotic distributions of sums of record values for distributions with regularly varying tails.
Let \(X_1,X_2,\dots\) be a sequence of independent and identically distributed random variables with continuous distribution function \(F\). It is assumed that \(F\) has a regularly varying tail, i.e., \(1-F(x)=x^{-\alpha}L(x)\) for \(x>0\), where \(\alpha>0\) and \(L(x)\) is a slowly varying function as \(x\to\infty\). For \(k\geq 2\), \(X_k\) is a record value if and only if \(X_k>\max\{X_1,\dots,X_{k-1}\}\). By convention, \(X_1\) is a record value. Let \(T_n\) be the sum of the first \(n\) record values. The main result of the paper says that, under the present assumptions, two sequences of positive numbers \(\{a_n\}\) and \(\{b_n\}\) with \(b_n\uparrow\infty\), \(n\to\infty\), exist such that \((\log(T_n)-\log(a_n))b_n^{-1}\) converges in distribution to the standard normal law, i.e., \((T_n/a_n)^{1/b_n}\) is asymptotically lognormal. A second result states that \(T_n\) cannot converge in distribution to any non-degenerate law through common centralizing and normalizing.
scientific article; zbMATH DE number 1897052