# Thread: iid uniform

1. ## 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 $n\to\infty$.

I appreciate any help.

2. 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 $n\to\infty$.

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)

3. you also need to show that

$E(\ln X)=\int_0^1 \ln xdx=-1$

4. Thanks. That I was able to prove.