Bulletin Editor Anirban DasGupta sets this problem. Student members of the IMS are invited to submit solutions (to bulletin@imstat.org with subject “Student Puzzle Corner”). The deadline is now April 15, 2016.

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We consider a problem on Gaussian extreme values. It comes across as a difficult calculation, but when looked at the right way, it is actually not at all difficult. Here is the exact problem.

Consider a sequence of iid random variables $X_1,X_2,\cdots \sim N(\mu ,{\sigma }^2)$. For any given $n\ge 1$, suppose $\overline{X}={\overline{X}}_n$ denotes the mean and $X_{(n)}$ denotes the maximum of the first $n$ observations $X_1,\cdots ,X_n$. Define ${\mu }_n(\overline{X})=E(X_{(n)}|\overline{X})$, and $V_n(\overline{X})=\text{Var}(X_{(n)}|\overline{X})$.

(a) Find explicit closed form deterministic sequences $a_n,b_n$ such that $b_n[{\mu }_n(\overline{X})-a_n]\stackrel{a.s.}{\to }1$.

(b) Find explicit closed form deterministic sequences $c_n,d_n$ such that $d_n[V_n(\overline{X})-c_n]\stackrel{a.s.}{\to }1$.

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The solution to the previous puzzle is here.