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January 26th, 2018, 10:25 PM  #1 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0  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. 
January 26th, 2018, 10:47 PM  #2 
Senior Member Joined: Oct 2009 Posts: 555 Thanks: 179 
What is $ln(S_n)$. Compute it.

January 26th, 2018, 11:06 PM  #3 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 
$\ln(S_n) =\frac{1}{n}\sum_{k=1}^n\ln(X_k)$, 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.

January 26th, 2018, 11:10 PM  #4 
Senior Member Joined: Oct 2009 Posts: 555 Thanks: 179 
This is where the law of large numbers comes in, which says something about limits like these.

January 26th, 2018, 11:13 PM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 
you'll probably need $\lim \limits_{n \to \infty} ~\left(1+\dfrac x n\right)^n = e^x$ 
January 27th, 2018, 07:11 AM  #6 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 
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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iid, limit, product, random, variables 
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