Convergence rates for the moments of extremes
Let \(X_1,\dots,X_n\) be independent copies of a random variable \(X\) such that NEWLINE\[NEWLINEM_n:=(\max_{1\leq i\leq n}X_i-b_n)/a_nNEWLINE\]NEWLINE converges in distribution with some norming sequences \(a_n>0\), \(b_n\in\mathbb R\), to a random variable \(\xi\), which follows an extreme value distribution. It is known that under appropriate conditions including \(E(|X|^k)<\infty\), one has \(\lim_{n\to\infty}E(M_n^k)=E(\xi^k)\) for an integer \(k>0\). The paper provides second order regularity conditions on the distribution function of \(X\), such that \((E(M_n^k)-E(\xi^k))/A(n)\) has a non degenerate limit as \(n\) tends to infinity, where \(A(\cdot)\) is an appropriate norming function.
scientific article; zbMATH DE number 6044474