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I'm working through an exercise related to a recent convergence/limit lecture for a probability theory course I'm taking and I seem to be doing something wrong, as I know the solution but seem to be getting off track at some point. The question:

Given an i.i.d. sequence $X_1,X_2...$ uniformly distributed on $[0,1]$, determine

$\lim_{n \to \infty} \sqrt[n]{X_1...X_n}$

With probability 1.

So, first I'm thinking that given the language "with probability 1" I should be thinking strong law of large numbers. So, with a sequence of uniform random variables on $[0,1]$, the variable converges a.s. to $\frac{1}{2}(a+b)$, or $\frac{1}{2}(0+1) = \frac{1}{2}$

Now, as n tends to infinity, the nth root of $\frac{1}{2}$ goes to 1.

The solution to the exercise is given as $e^{-1}$

Can anyone offer guidance on where I'm going wrong?

Thank you!

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The problem with your approach is that the Strong Law of Large Numbers is a statement about the arithmetic mean of $X_1,\dots,X_n$, while this problem is considering the geometric mean.

However, note that $$ \log(\sqrt[n]{X_1\cdots X_n})=\frac{\log(X_1)+\dots+\log(X_n)}{n} $$ and since the random variables $\{\log(X_n)\}_n$ are i.i.d., it is enough to determine their expected value, then appeal to the Strong Law of Large Numbers.

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  • $\begingroup$ Can't believe I didn't recognize that as a geo mean.. got it now thanks $\endgroup$ – Archetupon Nov 8 '16 at 23:57

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