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 $n \geq 1$, suppose $\bar{X} = \bar{X}_n$ denotes the mean and $X_{(n)}$ denotes the maximum of the first $n$ observations $X_1,\cdots , X_n$. Define $\mu_n(\bar{X}) = E(X_{(n)}|\bar{X})$, and $V_n(\bar{X}) =  \mbox{Var}(X_{(n)}|\bar{X})$.

(a) Find explicit closed form deterministic sequences $a_n, b_n$ such that $b_n[\mu_n(\bar{X})-a_n] \stackrel {a.s.} {\rightarrow} 1$.

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

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The solution to the previous puzzle is here.