Convergence of $V_n=\prod\limits_{i=1}^n U_i$

Question

I struggle to do this exercise:

Let $U_1,U_2,\dots$ be a sequence of i.i.d. random variables. We define $$V_n=\prod\limits_{i=1}^n U_i$$

Show that $V_n^{1/n}$ converges almost sure and calculate the limit.

So what I thought so far:

Since I have to show almost sure convergence, I would think that I have to show this with the law of large numbers.

So $$\lim\limits_{n\to \infty}\sum\limits_{i=0}^n\frac{\sqrt{V_n}}{n}=E[\sqrt{V_n}]=nE[\sqrt{U_i}]$$

Now here I'm stucked and I don't know if this is correct. Thanks for help.

Answer 1

Assuming that $U_i > 0$, we may define $X_i$ as $\log U_i$.
Assuming that $\mathbb{E}[\log U_i]<+\infty$, the law of large numbers gives $$ \frac{X_1+\ldots+X_n}{n}\to \mathbb{E}[\log U_i] $$ and by exponentiating back $$ V_n^{1/n} \to \exp\left(\mathbb{E}[\log U_i]\right). $$