# iid uniform

#### Veve

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.

#### Laurent

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

(Or equivalently take the logarithm of the sequence and apply the law of large numbers)

Veve

#### matheagle

MHF Hall of Honor
you also need to show that

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

Veve

#### Veve

Thanks. That I was able to prove.(Clapping)