# iid uniform

• May 20th 2010, 05:52 AM
Veve
iid uniform
Assume that $X_1,X_2,...,X_n$ are iid uniform on [0,1]. Show that $(X_1X_2...X_n)^\frac{1}{n}\to e^{-1}$ a.e. for $n\to\infty$.(Itwasntme)

I appreciate any help.
• May 20th 2010, 07:03 AM
Laurent
Quote:

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

I appreciate any help.

Express the power using $a^b=e^{b\log a}$ and use the law of large numbers.

(Or equivalently take the logarithm of the sequence and apply the law of large numbers)
• May 21st 2010, 10:02 PM
matheagle
you also need to show that

$E(\ln X)=\int_0^1 \ln xdx=-1$
• May 22nd 2010, 04:48 AM
Veve
Thanks. That I was able to prove.(Clapping)