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Let $X_1, ..., X_n \stackrel{iid}{\sim} N(0,1)$, define $\hat F_n(x) \equiv \frac{1}{n} \sum_{i=1}^n \mathbf{1}(X_i \leq x)$ and suppose $a_n$ is a sequence of (deterministic) numbers such that

$$a_ne^{\frac{a_n^2}{2}} = n \quad \quad (1)$$

I want to

(i) show that $\hat F_n(a_n) \xrightarrow{P} 1$ and

(ii) determine the limiting distribution of $n(1- \hat F_n(a_n))$.

I don't really have an idea of how to do this properly, but I've tried the following:

For (i): $a_n \rightarrow \infty$, for if it is bounded or goes to $-\infty$ we can't have the LHS in (1) equal to $n$. Now let $\epsilon > 0$ and choose $M \in \mathbb{R}$ such that $1-\Phi(M) < \epsilon$ where $\Phi(x) \equiv \int_{-\infty}^x \frac{e^{\frac{-x^2}{2}}}{\sqrt{2 \pi}}dx$. There must exist an $N$ such that $n \ge N \implies a_n > M$, in which case $1 \geq \hat F_n(a_n) \geq \hat F_n(M) \xrightarrow{a.s.} \Phi(M) > 1- \epsilon$

where the almost sure convergence is by Glivenko-Cantelli and then $\hat F_n(a_n) \xrightarrow{a.s.} 1$ which is even stronger than what I need. Does this work?

I have no idea where to start for (ii). Please help if you can!

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1 Answer 1

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What you did for the first part is correct.

For the second one, we have to find the limiting distribution of $Y_n:=\sum_{i=1}^n\mathbf 1\{X_i\gt a_n\}$. Using independence, the characteristic function of $Y_n$ is $$ \varphi_n(t)=\left(1+\left(e^{it}-1\right)\mathbb P\{X_1>a_n\}\right)^n.$$ Using the fact that $a_n\to+\infty$ and the lower and upper tail for Gaussian distribution, we find that $n\mathbb P\{X_1>a_n\}\to 1/\sqrt{2\pi}$. We can conclude using the fact that $(1+x_n/n)^n\to e^x$ if $x_n\to x\in (0,+\infty)$.

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  • $\begingroup$ Wouldn't the Gaussian tail bounds imply that $nP(X_1 > a_n) \rightarrow \frac{1}{\sqrt{2\pi}}$? $\endgroup$
    – qx123456
    Apr 18, 2020 at 17:29
  • $\begingroup$ Yes.${}{}{}{}{}$ $\endgroup$ Apr 18, 2020 at 17:31
  • $\begingroup$ Sorry to be asking you specifically so much, but you've been immensely helpful thus far. On a semi unrelated problem, would you know how to approach the following: math.stackexchange.com/questions/3631648/… $\endgroup$
    – qx123456
    Apr 18, 2020 at 19:02

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