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January 26th, 2018, 10:25 PM   #1
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Exclamation Limit of Product of iid Random Variables

Let $X_1, X_2, ... , X_n$ be i.i.d random variables with common density function:
$f(x)=\frac{\alpha}{x^{\alpha+1}}I(x\geq 1)$ where $I(x\geq 1)$ is the indicator function and $\alpha >0$. Now define:
$$S_n=\left[ \prod_{i=1}^{n}X_i \right ]^{n^{-1}}$$
Prove that $\lim_{n\rightarrow \infty}S_n=e^{\alpha^{-1}}$, and use the law of large numbers and the continuous mapping theorem to conclude about the convergence in probability of $ln(S_n)$.

I honestly don't have the slightest inkling on how to go about evaluating that limit. However, since the value is given in the problem statment, my intuition tells me that $ln(S_n)$ converges in probability to $\alpha^{-1}$, but I'm not certain.
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January 26th, 2018, 10:47 PM   #2
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What is $ln(S_n)$. Compute it.
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January 26th, 2018, 11:06 PM   #3
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$\ln(S_n) =\frac{1}{n}\sum_{k=1}^n\ln(X_k)$, but I still don't know how to compute the limit of $S_n$ as $n$ goes to infinity. Apparently I've forgotten part of calculus.
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January 26th, 2018, 11:10 PM   #4
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This is where the law of large numbers comes in, which says something about limits like these.
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January 26th, 2018, 11:13 PM   #5
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you'll probably need

$\lim \limits_{n \to \infty} ~\left(1+\dfrac x n\right)^n = e^x$
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January 27th, 2018, 07:11 AM   #6
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I understand how to do the problem using the strong law of large numbers, but I wasn't sure if I was supposed to prove the value of that limit some other way initially (the setup of the problem suggests so since it mentions so first).
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