Convergence rate for density of maximum of independent random variables (Q1382861)

Let \((X_n)_{n\geq 1}\) be a sequence of i.i.d. random variables with common distribution function \(F\) and put \(Z_n= \max(X_1,\dots, X_{n+ 1})\), \(n\geq 1\). Assume that there exist real numbers \(a_n\) and \(b_n> 0\), \(n\geq 1\), such that \(\lim_{n\to\infty} P((Z_n- a_n)/b_n< x)= H(x)\), \(x\in\mathbb{R}\), at all continuity points \(x\) of some nondegenerate distribution function \(H\). It is well-known that there are three types of possible limit distributions. Let \(f_n\) denote the probability density of \((Z_n- a_n)/b_n\), \(n\geq 1\), assuming that \(p(x)= F'(x)\), \(x\in\mathbb{R}\), exists. For any of the three possible types of \(H\), the authors give necessary and sufficient conditions for to have \(p_n(x)\to H'(x)\) as \(n\to\infty\) in the sense of locally uniform convergence.