The asymptotic distribution of sums of records
Let \(X_1,X_2,\dots\) be a sequence of independent identically distributed random variables. Denote by \(X^{(0)}=X_1, X^{(1)}, X^{(2)},\dots \) the associated upper record sequence and take \(T_n=\sum_{j=0}^n X^{(j)}\). The authors describe three cases in which \(T_n\) can be properly normalized to have a nontrivial limit distribution. More precisely, if \(X_1\) has the standard exponential or Gumbel distribution, then the sums of records are asymptotically normal; on the other hand, sums of beta records (including uniform records) are attracted to non-normal limit law \(L_{\gamma}\) with the Laplace transform \(\psi_{\gamma}(t)=\exp\{(-1/\gamma) \int_0^1 (1-\exp(-tx))/x dx\}\). These cases are investigated by essentially different approaches, and the problem of identifying all possible limit laws for the normalized \(T_n\) remains open.
scientific article; zbMATH DE number 1323316