Rate of convergence in the transference max-limit theorem (Q2769653)

Let \(X_j\), \(j\geq 1\), be a sequence of independent and identically distributed random variables. Let \(G_n(x)\), \(n\geq 1\), be a sequence of continuous and strictly increasing functions with domain containing the possible values of the \(X_j\). Define \(X_{nj}\) by \(G_n(X_{nj})= X_j\). Let \(N_n\) be a sequence of positive integer valued random variables, independent of the \(X_j\). Put \(M(n)\) and \(M(N_n)\), respectively, for the maximum of \(X_{nj}\), \(1\leq j\leq n\), and \(X_{nj}\), \(1\leq j\leq N_n\). Assume that \(N_n/n\) has a limiting distribution, and that a speed of convergence estimate is known for both \(N_n/n\) and \(M(n)\). By combining these estimates, the author develops a nonuniform speed of convergence for \(M(N_n)\). Earlier, the author developed similar results under the additional assumption that \(G_n(x)\) is linear, a standard assumption in the classical theory.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].