Convergence rates of limit distribution of maxima of lognormal samples (Q448264)

It is known that linearly normalized partial maxima of independent and identically distributed (iid) random variables (rvs) from the lognormal distribution converge weakly to a rv having the Gumbel distribution \(\Lambda.\) In this paper the authors prove the following result: If \(\{X_n, n \geq 1\}\) is a sequence of iid rvs with common lognormal distribution \(F\) and if the norming constants \(a_n\) and \(b_n\) are given by the solution of the equations NEWLINE\[NEWLINE2\pi (\log b_n)^2exp(\log b_n)^2=n^2,\;a_n=b_n/\log{b_n},NEWLINE\]NEWLINE then there exist absolute constants \(0< C_1 < C_2\) such that NEWLINE\[NEWLINEC_1/\sqrt{\log n} < \sup_{x \in R}\mid F^n(a_n x + b_n) - \Lambda(x)\mid <C_2/\sqrt{\log n},NEWLINE\]NEWLINE for \(n \geq 2\). And for the norming constants NEWLINE\[NEWLINE\alpha_n=exp{\sqrt{2 \log n}/\sqrt{2 \log n}}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\beta_n = \exp \sqrt{2 \log n}\left( 1 - \frac{\log 4\pi + \log \log n}{2 \sqrt{2 \log n}}\right),NEWLINE\]NEWLINE NEWLINE\[NEWLINEF^n(\alpha_n x + \beta_n) - \Lambda(x)\sim -\frac{e^{-x}\exp e^{-x}}{8} \frac{(\log \log n)^2}{\sqrt{2 \log n}}NEWLINE\]NEWLINE for large \(n\).