a.s. convergence uniform distribution

Question

I am having troubles with proving almost surely convergence for the following problem:

Let $U_j$ be IID $U(0,1)$ distributed and define $A_n$ to be:

$A_n=\sum_{k=1}^n \prod_{j=1}^k U_j$ for $n\in \mathbb{N}$.

for $n\rightarrow \infty$, I want to prove that $A_n$ converges almost surely to some $A$.

I tried with LLN, but this did not give anything.

Answer 1

Note that $A_n$ is an increasing sequence of random variables, and therefore the pointwise limit equals $$A=\sup_{n \in \mathbb{N}} A_n=\lim_{n \to \infty} A_n.$$ In order to show that this limit is well-defined (in the sense $A<\infty$ a.s.) we note that by the monotone convergence theorem

$$\mathbb{E} \left( \sup_{n \in \mathbb{N}} A_n \right) = \sup_{n \in \mathbb{N}} \mathbb{E}(A_n) = \sup_{n \in \mathbb{N}} \sum_{k=1}^n \frac{1}{2^k} < \infty,$$

and so $A \in L^1(\mathbb{P})$. This implies, in particular, $A<\infty$ almost surely.