[Math] Proof that the event that the series $\sum\limits_nX_n$ converges is in the tail sigma-field

measure-theoryprobability theory

This is probably very stupid question, but I'm stuck. Suppose that $(X_n)_n$ is a sequence of independent random variables. Let $\mathcal{F}_n = \sigma(X_n)$. I am trying to proof that

$\left\lbrace\omega\colon\sum_n X_n(\omega) < \infty\right\rbrace$

is a tail event. I suppose, that it should easily follow from the fact, that

$\left\lbrace\omega\colon\sum_{k=n}^{\infty} X_k(\omega) < \infty\right\rbrace\in\sigma(\mathcal{F}_n, \mathcal{F}_{n+1}, \ldots)$.

My problem is, I can't proof the last one. It makes me feel so stupid 🙁

Best Answer

Do it step by step.

  1. $X_n$ is $\sigma(X_n)$-measurable for each $n \in \mathbb{N}$. As $\mathcal{F}_n = \sigma(X_n)$, this is equivalent to saying that $X_n$ is $\mathcal{F}_n$-measurable.

  2. Since $\mathcal{F}_n \subseteq \mathcal{G}_n := \sigma(\mathcal{F}_n,\mathcal{F}_{n+1},\ldots)$, this implies that $X_n$ is $\mathcal{G}_n$-measurable.

  3. The sum $$\sum_{k=n}^N X_k$$ is, by step 2, $\mathcal{G}_n$-measurable as a finite sum of $\mathcal{G}_n$-measurable random variables.

  4. The limit (provided it exists) $$\sum_{k=n}^{\infty} X_k = \lim_{N \to \infty} \sum_{k=n}^N X_k$$ is $\mathcal{G}_n$-measurable as pointwise limit of $\mathcal{G}_n$-measurable random variables.

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