# Independence and almost sure convergence

Let $\{X_{n,i} : n \in \mathbb{N},i \in I\}$ be real-valued random variables on $(\Omega,\mathcal{F},\mathcal{P})$. Assume that for each $n \in \mathbb{N}$, the variables $\{X_{n,i} : i \in I\}$ are independent. Assume that for each $i \in I$ there is a real-valued random variable $Y_i$ on $(\Omega,\mathcal{F},\mathcal{P})$ such that $X_{n,i} \to Y_i$ a.s. as $n \to \infty$. How can I show that $\{Y_i$ : i $\in I\}$ are independent?

I tried to apply integral convergence theorem to random variables of the type $f(X_{n,i})$ where $f$ is a bounded, continuous function but I was stuck half way.

• Where exactly are you stuck?
– saz
Oct 15 '15 at 19:12
• I feel like the direction I head takes me no where. Oct 15 '15 at 19:15

Here is a criterion that might be useful: $(Y_i)_{i\in I}$ are (mutually) independent if and only if $$\Bbb E\left[\prod_if_i(Y_i)\right]=\prod_{i\in I}\Bbb E[f_i(Y_i)]$$ for all bounded continuous $f_i$.