# What is the tail $\sigma$-field?

In the book "probability : example and application", they define a tail $$\sigma$$-field as $$\mathcal T=\bigcap_{n=1}^\infty \mathcal F_n'$$ where $$\mathcal F_n'=\sigma (X_n,X_{n+1},...)$$. They call $$\mathcal t$$ remote future and say that $$A\in \mathcal T$$ if and only if changing a finite number of value do not affect the occurrence of the event.

I don't really understand what they want to say by this last sentence. Changing an infinite value of what ? of $$A$$ ? Could someone explain me better what represent $$\mathcal T$$ and in hat it is interesting ?

I have as example,

• If $$B_n\in \mathcal B(\mathbb R)$$, then $$\{X_n\in B_n\ \ i.o.\}\in \mathcal T$$, but I don't really understand why.

• Also, if $$S_n=X_1+...+X_n$$, then $$\{\lim_{n\to \infty }S_n\ exist\}\in \mathcal T$$ but $$\{\limsup_{n\to \infty }S_n>0\}\notin \mathcal T$$, and I also don't understand why.

• I am writing an answer, I request you to wait. – Teresa Lisbon Apr 12 '19 at 8:17

What you've got to think is that $$X_n$$ is the value of some random variable at time $$n$$. So, the "remote future" is what happens "after a very large time", that is, for $$n$$ very large. For convenience, I will assume my time unit as years, so $$X_0$$ is the value now, $$X_1$$ is the value after $$1$$ year, $$X_2$$ is the value after $$2$$ years (of some quantity, say price of a teddy bear at my favourite toy shop today is $$X_0$$, after $$1$$ year is $$X_1$$ , after two years is $$X_2$$ and so on).

Let us take examples to clarify this.

For example, consider $$\mathcal F_{23}'$$. This consists of events that don't depend on anything happening within $$23$$ years from now. So , you have absolutely no clue about this event for at least $$23$$ years from the start. However, it could depend on what happens after that.

Another way to say the same thing , but a way more convenient for usage : if I change the values $$X_1,...,X_{22}$$ to any value I like, the probability of any event in $$\mathcal F_{23}'$$ will be unaffected.

For example, the event : will $$X_{24}$$ be greater than $$76$$, is in $$F_{23} '$$, because it depends on something that doesn't happen within $$23$$ years of the start. You have no clue about $$X_{24}$$ until $$24$$ years, therefore certainly you will have to wait for at least $$23$$ years.

So is the event : will $$X_{23}$$ be bigger than $$X_{69}$$?

But the event $$X_{11} < X_{27}$$ won't be in $$F_{23}'$$ since we have $$X_{11}$$ here, which depends on something happening before $$23$$ years are up. After $$11$$ years, I will know the value of $$X_{11}$$ : so for example, if it is small, I will know that it is very likely that $$X_{27}$$ is going to be larger than it, so you have a clue about the event before $$23$$ years. Alternately, if I change $$X_{11}$$ to something very small, this probability increases, so this event can't be in $$\mathcal F_{23}'$$.

Similarly, the event $$X_7 + X_8 > 21 X_{29}$$ depend on values before $$23$$ years, so they won't be in $$\mathcal F_{23}'$$. If I change $$X_7,X_8$$ this probability is going to change, right?

Now, while it is slightly difficult to believe initially, it is definitely true that there are events, that don't depend on events happening before $$23$$ years, before $$24$$ years, before $$500$$ years, and so on.

These events are events that lie in every $$\mathbb F_n$$. In particular, they don't depend on anything any number of years from now, but instead depend on every possible distant future.

And with the changing interpretation, what you get is this : If I pick any $$X_{n_1},...,X_{n_k}$$ for any $$n_1,...,n_k \geq 1$$ and change their value, the event does not change.

For example, let us take an event from what you have, defined by $$\{X_n \in B_n i.o.\}$$.

Now, changing finitely many values of the $$X$$s is not going to affect this event, because that will reduce/increase the number of $$X_n$$ that are in $$B_n$$ by only a finite number, so the total number of $$X_n$$ still in $$B_n$$ will either remain infinite, or remain finite as per the initial configuration of the $$X_i$$.

For example, if I change say $$X_1,X_2,...,X_{2376}$$ to some values which are not in $$B_1,B_2,...,B_{2376}$$ respectively, then the event doesn't change, because if it were true, then $$X_n \in B_n$$ is happening for infinitely many $$n$$, and removing $$2376$$ of these is not going to affect the infiniteness.

Such events are part of what we call the tail sigma algebra. You have to think of the tail like the tail of a sequence : does the convergence of a sequence depend upon its first $$50,000$$ terms, for example? Its first $$10^{10^{10}}$$ terms? No. Properties which behave "like this" would be those in the tail sigma algebra.

Let us take the next example, $$\lim S_n$$ exists. Indeed, existence has nothing to do with the first finitely many terms, right? If $$S_n$$ existed, and I made all of $$X_1,...,X_{2354}$$ very large, then the sum will still exist. If $$S_n$$ did not exist, and I made all of $$S_{45},S_{24},S_{456456}$$ small, it is not going to help : the sum still won't exist.

That makes it a tail event.

Now, why is the event $$\limsup S_n > 0$$ not a tail event? Well, because unfortunately , when you take a sum of terms, every non-zero term contributes, and unfortunately the largest term contributes the most. So, for example, imagine that I had $$\limsup S_n <0$$. By making $$X_1$$ very very large, each of $$S_1,S_2,...$$ includes $$X_1$$ in its sum, so it is possible that I can lift the limit superior of $$S_n$$ over zero by taking $$X_1$$ sufficiently large.

Look at this from the series point of view. Suppose you have an infinite series which sums to $$-1$$ which is less than $$0$$. By increasing the first term by $$2375$$ , I will make the sum $$2374$$, which is greater than $$0$$.

So the sum being negative is not a tail event, because increasing /decreasing a few finitely many terms (above, I changed only the first term $$X_1$$) can potentially make it negative/positive!

Naturally the most important result regarding the tail sigma algebra has to be Kolmogorov's theorem : if $$X_i$$ are independent events, then any event in the sigma algebra has probability zero or one.

So what makes the tail sigma algebra so powerful is the assertion that any event belonging in it has an easy to find probability : if you can show it has non-zero probability, then it has probability one.

For example, if $$X_n$$ are independent , then the tail sigma events we discussed above would have probability zero or one (which one? That depends upon what the $$X_n$$ are, of course).

• wwwaaaouuuuuu, that's an answer ! Thanks a lot. I don't have time to read it now, but I'll give you a feed back soon. Thanks a lot :) – user659895 Apr 12 '19 at 17:10
• By the way, you really look to be an expert in probability... could you have a check to this question ? Maybe you'll have an idea :) – user659895 Apr 12 '19 at 17:11
• I do have a clue about that question! But unfortunately I have to go to sleep now, so I will get back as soon as I can regarding that – Teresa Lisbon Apr 12 '19 at 17:36

Consider the following events:

a) $$\{\sup X_n <1\}$$

b) $$\sum X_n \, \text {is convergent}\}$$

c) $$\{\lim \sup X_n <1\}$$

The events in b) and c) are tail events but the one in a) is not. The condition $$\sup x_n <1$$ (for a sequence of real numbers $$(x_n)$$ depends on every term of the sequence: it may hold for one sequence but it may not holds when the first term of the sequence is altered. On the other hand the condition $$\lim \sup X_n <1$$ does not change if you change the first $$100$$ or the first $$1000$$ terms. Similar thing is true about the condition $$\sum x_n$$ convergent.

• Thank you, but at the beginning my problem is to understand what is $\mathcal T$ and why is it important ? For me, it doesn't look interesting, because no $X_n$ are $\mathcal T$ measurable... by the way $\mathcal T$ looks empty... (for example $\bigcap_{n\geq 1}\{n,n+1,...\}=\emptyset$... here it looks to be the same thing...) – user659895 Apr 12 '19 at 8:19
• Sets in $\mathcal T$ need not be empty. For example the event in b) is a tail event and it is not empty. If $X_n=X$ for all $n$ the the event in c) is the entire sample space. The importance of $\mathcal T$ comes from the fact that when $X_i$'s are independent the sets in $\mathcal T$ have probability $0$ or $1$ and this is very useful in probability theory. – Kavi Rama Murthy Apr 12 '19 at 8:25

$$\{X_n \in B_n \text{ i.o}\} = \bigcap_n\bigcup_{m \geq n} \{X_m \in B_m\}$$

Note that $$\bigcup_{m \geq n} \{X_m \in B_m\} \in \mathcal{F}_n'$$.

Therefore, $$\{X_n \in B_n \text{ i.o}\} \in \mathcal{F}_n'$$ for every $$n$$. Hence, it belongs to the tail sigma-algebra.