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Show $\lim X_k < \infty$ is in tail sigma-algebra

Given random variables $X_1, X_2, X_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(X_{n+1}, X_{n+2}, ...)$ be their tail sigma-algebra.

For convenience, $\tau_n \doteq \sigma(X_{n+1}, X_{n+2}, ...)$.

What I tried:

$\forall n \in \mathbb{N}$,

$(\lim X_k < \infty) = (\lim X_{k+n} < \infty)$

$\forall n \in \mathbb{N}, (\lim X_k < \infty) = (\lim X_{k+n} < \infty)$

$\to (\lim X_k < \infty) \in \tau_n \forall n \in \mathbb{N} \because$

$X_{n+1}, X_{n+2}, ...$ are RVs on $(\Omega, \tau_n, \mathbb{P})$ --> Is this right?

If so, then obviously $(\lim X_k < \infty) \in \tau$

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1 Answer 1

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If you want to know whether $\lim_{n\rightarrow\infty}X_n<\infty$ then you can just pick out some $k\in\mathbb N$ and have a look at sequence $X_{k+1},X_{k+2},\dots$.

The values taken by $X_1,\dots,X_k$ are simply irrelevant when it comes to this question, and this is the case for any $k\in\mathbb N$.

This observation allows the conclusion that event $\{\lim_{n\rightarrow\infty}X_n<\infty\}$ belongs to tail $\sigma$-algebra $\tau$.

This reasoning works in many other cases (concerning e.g. $\limsup$, $\liminf$, summation et cetera)

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  • $\begingroup$ k instead of n? Thanks but how about my reasoning? $\endgroup$
    – BCLC
    Commented May 23, 2015 at 4:24
  • $\begingroup$ your reasoning is okay. $\endgroup$
    – drhab
    Commented May 23, 2015 at 7:28
  • $\begingroup$ why this is enough to conclude? $\endgroup$ Commented Apr 14 at 1:22
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    $\begingroup$ @FranAguayo Formally it must be shown that for every fixed $k$ the event $\lim_{n\to\infty}X_n<\infty$ is an element of the $\sigma$-algebra $\tau_k$ that is generated by the rv's $X_n$ with $n>k$. This can be done (I do not provide the details of this in my answer) and that means that the event is an element of the tail $\sigma$-algebra $\tau=\bigcap_{k=1}^{\infty}\tau_k$. $\endgroup$
    – drhab
    Commented Apr 14 at 14:32
  • $\begingroup$ thanks for the answer! $\endgroup$ Commented Apr 15 at 1:43

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