Understanding terminal $\sigma-$algebra and related notation

Question

I need some clarification about this "statement" in my notes, which I find unclear and I could not find anything really comprehensive around. I am studying for an exam so it's important.

"Be $(X_n)$ a sequence of random variables. Be $(\mathcal{G}_n) := \mathcal{F}(X_1, \ldots, X_n)$, and $\mathcal{G}_n$ is not increasing. $$\mathcal{G} := \bigcap_n \mathcal{G}_n$$ is called terminal $\sigma-$ algebra of $(X_n)$."

My confusions about this are the following:

• I don't understand this notation: $\mathcal{F}(X_1, \ldots, X_n)$. Is this a sort of "the sigma algebra generated by all those random variables"? Even if that were the case, I don't understand the meaning.

• Why we require $\mathcal{G}_n$ not increasing?

• I cannot find a good definition / explanation of what a terminal $\sigma-$ algebra means in "concrete". What is its purpose? Why we need it?

Thank you!!

Answer 1

Curiously, the wikipedia page for Kolmogorov 0-1 also refers to this object as the "terminal sigma algebra". The more commonly used term is the "tail sigma algebra".

It is obtained as follows. Denote $$T_n := \sigma (X_n,X_{n+1},\ldots) = \sigma \left(\{X_k^{-1}(B) \mid B\in\mathcal B(\mathbb R), k\geqslant n\} \right),\quad n\in\mathbb N.$$ The intersection $T := \bigcap_{n\in\mathbb N} T_n$ is called the tail sigma algebra.

Tail events, i.e elements of $T$, are those whose outcome does not depend on any finite selection of the given variables. In other words, the beginning of the sequence does not matter. Kolmogorov 0-1 states that if we have a sequence of independent variables, then any tail event has probability of either $0$ or $1$.

Naturally, $\emptyset,\Omega\in T$. Other examples of tail events include the following:

1. $\exists \lim X_n := \{\omega \mid \exists \lim X_n(\omega)\}$
2. $\limsup X_n<\infty := \{\omega \mid \limsup X_n(\omega) < \infty\}$

..and so on. Whatever you can think of that makes use of upper\lower limits qualifies.