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Having a little trouble with this one:

Suppose $X_1, X_2, \ldots $ are iid standard normal random variables. Let $W_n = \sqrt{n} \frac{X_1 + \cdots + X_n}{X_1^2 + \cdots + X_n^2}$. Find the limiting distribution of $W_n$ as $n \to \infty$.

Too bad convergence in distribution isn't closed under division. Can't get slutsky's to apply.

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3 Answers 3

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Notice that $\frac{X_1 + X_2 + \ldots+ X_n}{\sqrt n}$ is a a standard gaussian variable. Then apply Slutsky's theorem.

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  • $\begingroup$ $\frac{\sum (x_i^2)}{n}$ converges in probability by WLLN? $\endgroup$
    – Taylor
    Jan 3, 2013 at 13:45
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The numerator is normally distributed with mean $0$ and variance $n$, and the denominator is distributed $\chi^2$ with $n$ degrees of freedom. Start there.

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We can find the result with law of large numbers and central limit theorem, writing $$W_n=\color{blue}{\frac n{\sum_{j=1}^nX_j^2}}\cdot\color{red}{\frac 1{\sqrt n}\sum_{j=1}^nX_j}.$$ The blue term converges to $E(X_1^2)^{-1}$ in probability by the law of large numbers.

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    $\begingroup$ (+1) Welcome, Davide. As you know, implicitly, we are using what is often known as Slutsky's Theorem here. I hope you have a look around and continue to participate. We occasionally get some fun probability questions here that don't appear on math.SE. For the future, combining red and green may not be the best option. $\endgroup$
    – cardinal
    Jan 6, 2013 at 16:13
  • $\begingroup$ Re the first term math.stackexchange.com/questions/2578090/… $\endgroup$
    – sjw
    Jul 9, 2020 at 23:30

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