The index of the outstanding observation among n independent ones (Q1098189)

Consider a sequence of independent but not identically distributed random variables. \textit{D. Mejzler} [Lvov. Politehn. Inst. Naucn. Zap. Ser. Fiz.- Mat. 38, 90-109 (1956)] made a thorough study of the possible limiting forms of the maximum \(M_ m\) in those sequences. Associated with the maximum is also the index \(L_ n\) where the maximum is attained for the first time, both \(M_ n\) and \(L_ n\) considered in the sample of the first n observations. If \(M_ n\), conveniently reduced, converges to one of the possible limiting distributions then the limiting distribution of \(L_ n/n\) is obtained, and expressed functionally in the limiting distribution of the reduced \(M_ n\). Some useful results follow. The converse problem of obtaining the limiting distribution of the reduced \(M_ n\) when the limiting distribution of \(L_ n/n\) is known is raised and partially solved, even when we consider a more general reduction of \(L_ n\).