Assume that $\displaystyle X_1,X_2,...,X_n$ are iid uniform on [0,1]. Show that $\displaystyle (X_1X_2...X_n)^\frac{1}{n}\to e^{-1}$ a.e. for $\displaystyle n\to\infty$.(Itwasntme)

I appreciate any help.

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- May 20th 2010, 04:52 AMVeveiid uniform
Assume that $\displaystyle X_1,X_2,...,X_n$ are iid uniform on [0,1]. Show that $\displaystyle (X_1X_2...X_n)^\frac{1}{n}\to e^{-1}$ a.e. for $\displaystyle n\to\infty$.(Itwasntme)

I appreciate any help. - May 20th 2010, 06:03 AMLaurent
- May 21st 2010, 09:02 PMmatheagle
you also need to show that

$\displaystyle E(\ln X)=\int_0^1 \ln xdx=-1$ - May 22nd 2010, 03:48 AMVeve
Thanks. That I was able to prove.(Clapping)