Tail Sigma-Algebras of Independent Sequences

Why Tail Information Is Different

A sequence may have properties that depend on its long-run behavior rather than on any particular finite list of its first values. For example, whether a sequence visits a specified set infinitely often cannot be determined by inspecting only its first ten terms. Such properties are described using the tail sigma-algebra.

For an independent sequence, a key fact is that a tail event is independent of every fixed finite initial segment. The event may involve the entire future of the sequence, but its probability does not become associated with the first \(n\) coordinates. We will define this precisely and prove the assertion.

Definition of the Tail Sigma-Algebra

Let \((\Omega,\mathcal F,\mathbb P)\) be a probability space, and let \(X_1,X_2,\ldots\) be a sequence of random elements. For \(m\in\mathbb N_0\), define the sigma-algebra generated by the sequence after its first \(m\) terms by

\mathcal G_m=\sigma(X_{m+1},X_{m+2},\ldots).

Thus \(\mathcal G_0\) is generated by the whole sequence, while \(\mathcal G_m\) records only the coordinates after \(X_m\). Each successive tail sigma-algebra is contained in the preceding one: \(\mathcal G_{m+1}\subseteq\mathcal G_m\).

Definition: The tail sigma-algebra of the sequence \((X_n)_{n\ge1}\) is

\mathcal T=\bigcap_{m\in\mathbb N_0}\mathcal G_m =\bigcap_{m\in\mathbb N_0}\sigma(X_{m+1},X_{m+2},\ldots).

An event \(A\in\mathcal T\) is called a tail event.

Membership in \(\mathcal T\) means that the event belongs to every \(\mathcal G_m\), not merely to the sigma-algebra generated by the whole sequence. In particular, for each \(m\), the event must be expressible using coordinates strictly after \(m\). The descriptions may differ with \(m\).

Proposition: The tail sigma-algebra \(\mathcal T\) is a sigma-algebra.

Proof. Each \(\mathcal G_m\) is a sigma-algebra. Their intersection contains \(\Omega\), since \(\Omega\in\mathcal G_m\) for every \(m\). If \(A\in\mathcal T\), then \(A\in\mathcal G_m\) for every \(m\), so \(A^c\in\mathcal G_m\) for every \(m\), and hence \(A^c\in\mathcal T\). If \(A_1,A_2,\ldots\in\mathcal T\), then all the \(A_j\) belong to each \(\mathcal G_m\). Closure of each \(\mathcal G_m\) under countable unions gives \(\bigcup_{j=1}^\infty A_j\in\mathcal G_m\) for every \(m\), and therefore \(\bigcup_{j=1}^\infty A_j\in\mathcal T\). Thus \(\mathcal T\) is a sigma-algebra. \(\square\>

Worked Examples of Tail Events

Worked Example: Infinitely Many Visits to a Set

Let \(E_k\in\mathcal F\) be an event determined by \(X_k\), and consider the event that \(E_k\) occurs infinitely often:

A=\limsup_{k\to\infty}E_k =\bigcap_{q=1}^{\infty}\ \bigcup_{k=q}^{\infty}E_k.

Fix \(m\in\mathbb N_0\). The event can also be written

A=\bigcap_{q=m+1}^{\infty}\ \bigcup_{k=q}^{\infty}E_k.

Indeed, \(A\) means that for every starting index \(q\), some \(E_k\) occurs at an index \(k\ge q\). Restricting the starting indices to \(q\ge m+1\) does not change that condition: it only removes finitely many possible starting indices. Every event on the right is built from \(E_k\) with \(k\ge m+1\), so it belongs to \(\mathcal G_m\). This holds for every \(m\), and hence \(A\in\mathcal T\).

Worked Example: Eventually Staying in a Measurable Set

Suppose each \(X_k\) takes values in a measurable space \((S,\mathcal S)\), and let \(C\in\mathcal S\). Consider the event that \(X_k\in C\) for all sufficiently large \(k\):

B=\bigcup_{q=1}^{\infty}\ \bigcap_{k=q}^{\infty}\{X_k\in C\}.

For any fixed \(m\), this event also has the representation

B=\bigcup_{q=m+1}^{\infty}\ \bigcap_{k=q}^{\infty}\{X_k\in C\}.

If the condition holds from some index \(q\), it also holds from any larger index, so a suitable \(q\) can always be chosen at least \(m+1\). Conversely, such a \(q\) is certainly a valid starting index for the original definition. The displayed representation uses only coordinates after \(m\); thus \(B\in\mathcal G_m\). Since \(m\) was arbitrary, \(B\in\mathcal T\).

Worked Example: Infinitely Many Occurrences of a Fixed Block

Suppose the \(X_k\) take values in a measurable space \((S,\mathcal S)\). Fix a block of measurable sets \(C_1,\ldots,C_r\in\mathcal S\), where \(r\in\mathbb N\). Define

H_k=\bigcap_{j=1}^{r}\{X_{k+j-1}\in C_j\}.

Thus \(H_k\) says that the specified block of conditions occurs starting at position \(k\). The event that it occurs infinitely often is

D=\bigcap_{q=1}^{\infty}\ \bigcup_{k=q}^{\infty}H_k.

For a fixed \(m\), replace the outer intersection by one over \(q\ge m+1\). This does not change \(D\), because infinitely many starting positions remain after any finite index. In the resulting expression, every \(H_k\) uses only coordinates with indices at least \(m+1\). Therefore \(D\in\mathcal G_m\) for every \(m\), so \(D\in\mathcal T\). The block can overlap itself, and no disjointness assumption is needed for this measurability argument.

Independence from a Finite Initial Segment

For \(n\in\mathbb N\), the sigma-algebra generated by the first \(n\) coordinates is

\mathcal F_n^{X}=\sigma(X_1,\ldots,X_n).

Independence of sigma-algebras means that every event in one is independent of every event in the other: sigma-algebras \(\mathcal A\) and \(\mathcal B\) are independent if \(\mathbb P(C\cap D)=\mathbb P(C)\mathbb P(D)\) for all \(C\in\mathcal A\) and \(D\in\mathcal B\).

We use the established independence-of-infinite-families result: if the random elements in a sequence are independent, then sigma-algebras generated by disjoint sets of indices are independent. In particular, for each fixed \(n\), \(\sigma(X_1,\ldots,X_n)\) and \(\sigma(X_{n+1},X_{n+2},\ldots)\) are independent.

Theorem: Suppose \(X_1,X_2,\ldots\) is an independent sequence. For every \(n\in\mathbb N\), every tail event \(A\in\mathcal T\), and every \(B\in\sigma(X_1,\ldots,X_n)\),

\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B).

Equivalently, each tail event is independent of the sigma-algebra generated by any fixed finite initial segment.

Proof. Fix \(n\in\mathbb N\), \(A\in\mathcal T\), and \(B\in\sigma(X_1,\ldots,X_n)\). By the definition of the tail sigma-algebra, \(A\in\mathcal G_n=\sigma(X_{n+1},X_{n+2},\ldots)\). Independence of the sequence implies that \(\mathcal F_n^{X}\) and \(\mathcal G_n\) are independent. Since \(B\in\mathcal F_n^{X}\) and \(A\in\mathcal G_n\), the definition of independence gives

\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B).

The choices of \(n\), \(A\), and \(B\) were arbitrary, proving the assertion. \(\square\)

Worked Example: Infinitely Many Successes in Bernoulli Trials

Worked Example: Independence from the First \(n\) Trials

Let \(X_1,X_2,\ldots\) be independent Bernoulli random variables with \(\mathbb P(X_k=1)=\mathbb P(X_k=0)=1/2\), and let \(A\) be the event that \(X_k=1\) infinitely often. The first worked example shows that \(A\in\mathcal T\). Take \(B=\{X_1=1,\ldots,X_n=1\}\). Independence of the trials gives \(\mathbb P(B)=2^{-n}\). The theorem therefore yields

\mathbb P(A\cap\{X_1=1,\ldots,X_n=1\}) =\mathbb P(A)\,2^{-n}.

The conclusion concerns the joint probability of \(A\) and this particular initial pattern; it does not require calculating \(\mathbb P(A)\). More generally, the theorem applies to every event determined by the first \(n\) trials, not only to this one pattern.

What the Result Does—and Does Not—Say

The theorem is a finite-prefix statement. For any chosen \(n\), the event \(A\) is measurable using coordinates after \(n\), and independence separates it from the first \(n\) coordinates. The independence assumption is essential to this conclusion; tail measurability alone does not assert independence.

It is also important to keep the quantifiers straight. The theorem says that for each fixed \(n\), \(A\) is independent of every event in \(\sigma(X_1,\ldots,X_n)\). It does not, by itself, state independence from an arbitrary sigma-algebra containing all coordinates. That distinction keeps the result here focused on finite initial segments.

Key takeaway: A tail event belongs to the sigma-algebra generated by the coordinates after every finite cutoff. For an independent sequence, choosing the cutoff at \(n\) makes that event independent of the first \(n\) coordinates.

Check Your Understanding

Use the definitions and theorem to answer the following questions.

What does it mean for an event to belong to the tail sigma-algebra?
Why does removing finitely many possible starting indices not change an infinitely-often event?
For an independent sequence, which two sigma-algebras are used to prove independence from the first \(n\) coordinates?
Why is an event that eventually satisfies \(X_k\in C\) measurable with respect to every \(\mathcal G_m\)?
Does the theorem assert independence from any sigma-algebra that contains all the coordinates? Explain which independence statement it proves directly.