I am solving problem 9.4.3. from Theory of Probability and Random Processes (Koralov, Sinai). The problem says the following:

Let $\xi_1,\xi_2,\dots$ be a sequence of r.v. on a probability space $(\Omega, \mathcal{F}, P)$, and $\mathcal{G}$ be a $\sigma$-subalgebra of $\mathcal{F}$. Assume that $\xi_n$ is independent of $\mathcal{G}$ for each $n$, and that $\xi_n \to \xi$ almost surely. Prove that $\xi$ is independent of $\mathcal{G}$.

What I have done is the following: For each $A \subset \mathcal{G} $ take $\mathbb{1}_A$ the indicator function. Now I use the fact that $\xi_n$ and $\mathbb{1}_A$ are independent and so the characteristic function of the joint probability is the product of characteristic functions. I take the limit in both sides and therefore $\xi$ must be also independent of $\mathbb{1}_A$ for each $A$.

Now, am I correct in doing that? I think I use almost surely convergence of $\xi_n$ when I take the limit in the joint characteristic function, am I right? Or am I making some mistake here?

Thank you very much for your help.


Yes, your argumentation is correct. Note that the equality

$$\lim_{n \to \infty} \mathbb{E}\exp \big(\imath \, \eta_1 1_A+ \imath \eta_2 \xi_n\big) = \mathbb{E}\exp \big(\imath \, \eta_1 1_A + \imath \eta_2 \xi\big), \qquad \eta_1,\eta_2 \in \mathbb{R},$$

holds by the dominated convergence theorem (i.e. we can interchange limit and expectation).


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