Permutation-Invariant Events Versus Tail Events

Two Ways an Event Can Ignore Early Coordinates

An event may be insensitive to changes in finitely many coordinates in more than one sense. A tail event is determined by the coordinates after every finite cutoff. An event invariant under finite permutations is unchanged when finitely many coordinates are rearranged. These ideas are related, but they are not equivalent.

The distinction matters when applying zero–one laws. For independent random variables, tail events fall under Kolmogorov’s zero–one law. For an independent identically distributed sequence, the Hewitt–Savage zero–one law applies to events invariant under finite permutations. Understanding which class an event belongs to helps identify the right theorem without mistaking one kind of invariance for another.

Definitions on Sequence Space

Let $(S,\mathcal S)$ be a measurable space and consider the product space $S^{\mathbb N}$ with its product sigma-algebra. Write $x=(x_1,x_2,\ldots)$ for a point of this space, and let $X_n(x)=x_n$ denote its $n$th coordinate. The same definitions apply to an event about the paths of a random sequence by taking its inverse image under the map $x\mapsto (X_1(x),X_2(x),\ldots)$.

Definition: For $m\in\mathbb N$, the sigma-algebra generated by coordinates from $m$ onward is $\sigma(X_m,X_{m+1},\ldots)$. The tail sigma-algebra is $$ \mathcal T=\bigcap_{m=1}^{\infty}\sigma(X_m,X_{m+1},\ldots). $$ An event $A$ is a tail event if $A\in\mathcal T$. Equivalently, for every $m$, its membership is determined by the coordinates $x_m,x_{m+1},\ldots$.

A finite permutation is a permutation $\pi:\mathbb N\to\mathbb N$ that moves only finitely many indices. It acts on sequences by rearranging coordinates: $(\pi x)_n=x_{\pi^{-1}(n)}$. An event is invariant under finite permutations if rearranging a sequence in this way never changes whether it belongs to the event.

Definition: An event $A\subseteq S^{\mathbb N}$ is invariant under finite permutations if, for every finite permutation $\pi$ and every $x\in S^{\mathbb N}$, $$ x\in A \quad\Longleftrightarrow\quad \pi x\in A. $$

These definitions describe events as sets of paths, so invariance here is an exact, pointwise property. In probability applications, an event about a random sequence is obtained by pulling back such a set along the sequence-valued map. This makes clear which transformations of paths are being considered.

Every Tail Event Is Permutation-Invariant

A finite permutation only rearranges a finite set of coordinates. A tail event can be tested using coordinates beyond that set, where the rearranged sequence is unchanged.

Theorem: Every tail event on $S^{\mathbb N}$ is invariant under finite permutations.

Proof. Let $A\in\mathcal T$, and let $\pi$ be a finite permutation. Choose $m$ larger than every index moved by $\pi$. Then $(\pi x)_n=x_n$ for every $n\ge m$. Since $A\in\sigma(X_m,X_{m+1},\ldots)$, membership in $A$ depends only on these coordinates. Thus $x\in A$ if and only if $\pi x\in A$. Since this holds for every finite permutation, $A$ is invariant under finite permutations. $\square$

The reverse implication does not hold in general. Invariance permits rearranging coordinates, but it does not require an event to be unaffected by changing their values. A tail event must ignore any finite collection of initial values altogether. This difference is the reason that a permutation-invariant event need not be a tail event.

Worked Example: A Permutation-Invariant Event That Is Not a Tail Event

Worked Example: At Least One Coordinate Equals One

Work on $\{0,1\}^{\mathbb N}$, and let $A=\{x:\text{there exists }n\in\mathbb N\text{ with }x_n=1\}$. A permutation changes only the locations of the coordinate values, not whether at least one value is $1$. Therefore $A$ is invariant under every finite permutation.

To check that $A$ is not a tail event, compare the all-zero sequence $x=(0,0,\ldots)$ with $y=(1,0,0,\ldots)$. For every $m\ge2$, the two sequences agree in every coordinate from $m$ onward. But $x\notin A$ and $y\in A$. An event determined by the coordinates from $m$ onward could not distinguish them, so $A\notin\sigma(X_m,X_{m+1},\ldots)$. In particular, $A\notin\mathcal T$.

Under an independent identically distributed law, the Hewitt–Savage zero–one law says that a measurable event invariant under finite permutations has probability zero or one. The event in this example meets that invariance criterion, although it is not a tail event. By contrast, Kolmogorov’s zero–one law concerns tail events of an independent sequence. The laws have related domains, but the domains are not the same.

Worked Example: Empirical Frequency

Worked Example: The Limiting Frequency of Ones

Again let $x\in\{0,1\}^{\mathbb N}$, and consider $$ A_q=\left\{x:\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}x_j=q\right\}, $$ where $q\in\mathbb R$. The event is measurable because it is defined by convergence of a sequence of measurable finite-coordinate averages and by the value of its limit.

If two sequences differ only in their first $r$ coordinates, their averages through $n$ differ by at most $r/n$. This bound tends to zero, so one sequence has limiting frequency $q$ if and only if the other does. More generally, for each cutoff $m$, deleting the first $m-1$ terms does not change the limiting frequency: the averages of the remaining terms, taken with their own sample size, have the same limit. Thus membership is determined by coordinates from $m$ onward for every $m$, and $A_q$ is a tail event.

A finite permutation also changes only finitely many entries, so the same bound shows that the limiting frequency, when it exists, is unchanged. Hence $A_q$ is permutation-invariant as well. This event belongs to both classes; it does not demonstrate that the classes coincide.

Worked Example: Convergence and the Coordinate Limit

Worked Example: A Real Sequence Has a Limit

On $\mathbb R^{\mathbb N}$, let $C$ be the event that $x_n$ converges to a finite real number. Convergence is determined by the Cauchy condition: for every $\varepsilon>0$, there is an $N$ such that $|x_j-x_k|<\varepsilon$ whenever $j,k\ge N$. Changing or rearranging finitely many terms does not change whether this condition holds, since $N$ can be chosen beyond all affected indices. Therefore $C$ is a tail event and is invariant under finite permutations.

For any fixed $a\in\mathbb R$, the event that the sequence converges to $a$ has the same two properties. It can be expressed by requiring that for every $\varepsilon>0$, eventually $|x_n-a|<\varepsilon$. Both eventual conditions are unchanged by altering or rearranging a finite number of coordinates.

For a real-valued random sequence, the limit on the event of convergence is therefore determined by the tail. One may define it to be $0$ off the convergence event; this defines a tail-measurable random variable. This example illustrates that a limit event is not merely permutation-invariant: it is also tail because changing finitely many terms cannot affect eventual behavior.

Finite-Coordinate Statistics Usually Belong to Neither Class

A statistic based on a fixed finite block can depend on the particular values and locations in that block. That generally prevents it from being a tail event, and a permutation that moves a value into or out of the block can change it. Such a statistic should not be confused with a symmetric statistic of the entire infinite sequence.

Worked Example: A Statistic of the First Coordinate

On $\{0,1\}^{\mathbb N}$, let $B=\{x:x_1=1\}$. This is a finite-coordinate event. It is not a tail event: the all-zero sequence and the sequence with $x_1=1$ and all later coordinates zero agree from coordinate $2$ onward, but only the second belongs to $B$.

It is also not invariant under finite permutations. Take $x=(1,0,0,\ldots)$ and swap coordinates $1$ and $2$. The resulting sequence begins $(0,1,0,\ldots)$, so the original sequence belongs to $B$ and the rearranged one does not. Thus $B$ belongs to neither class.

This is a typical outcome, not a universal rule for every finite-coordinate statistic. A statistic may be constant, for example, and then its event sets are invariant and tail. The correct test is always to check the definitions: does membership depend only on every tail, and is it unchanged by every finite permutation?

Why the Distinction Matters

For an independent sequence, Kolmogorov’s zero–one law applies to events in the tail sigma-algebra. The Hewitt–Savage zero–one law applies when the sequence is independent and identically distributed and the event is invariant under finite permutations. Since every tail event is permutation-invariant, the latter theorem can cover tail events in the identically distributed setting; its additional reach is that it also covers invariant events that are not tail events.

A common error is to assume that rearranging finitely many coordinates and changing finitely many coordinate values are interchangeable operations. They are not. A tail event must be unaffected by the values of any finite initial block. A permutation-invariant event may depend on those values, provided it does not depend on their labels or positions. The event “some coordinate equals one” makes this contrast explicit.

When classifying an event, first identify its underlying sequence-space description. Then test tail measurability by asking whether its membership is determined after every cutoff. Separately test permutation invariance by applying a finite rearrangement. If the event is invariant and the coordinates are iid, the Hewitt–Savage theorem is available; if it is tail and the coordinates are independent, Kolmogorov’s theorem is available. Neither classification should be inferred solely from how an event is informally described.

Check Your Understanding

For each question, distinguish carefully between ignoring finitely many values and ignoring their positions.

Why does every tail event remain unchanged under a finite permutation?
Why is the event that at least one coordinate equals one permutation-invariant but not a tail event?
Is the event that the limiting frequency of ones equals a specified value a tail event, a permutation-invariant event, both, or neither? Explain.
Why are convergence of a real sequence and convergence to a fixed real number unchanged by finite permutations?
For the event $X_1=1$, give one pair of sequences showing it is not a tail event and one finite permutation showing it is not invariant.