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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.

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

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  • $\begingroup$ It should not be for all bounded functions? $\endgroup$
    – lele
    May 16 '16 at 18:32
  • $\begingroup$ ah, ok, you can think in characteristic functions... $\endgroup$
    – lele
    May 16 '16 at 18:37

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