nLab tail event

Idea

In probability theory, a tail event is an event which, intuitively, does not depend on the initial terms of a sequence. (For example, think of the truth value of the statement “The sequence $(x_n)$ has a limit”: it does not depend on the first terms, but only on the “tail” of the sequence.)

Sometimes, in fields such as economics?, tail event refers to the mostly unrelated notion of rare event. (Think of a point very far right or very far left of a gaussian distribution?, i.e. on the “tail”.)

Definition

Let $(X,\mathcal{A})$ be a measurable space. Consider the countably infinite product $X^\mathbb{N}$, and recall that the product sigma-algebra is given by

$$\sigma \left( \bigcup_{i\in\mathbb{N}} \pi_i^{-1}(\mathcal{A}) \right) ,$$

where $\pi_i:X^\mathbb{N}\to X$ is the $i$-th product projection.

The tail sigma-algebra on $X^\mathbb{N}$ is the sub-sigma-algebra of the product one, given by

$$\bigcap_{n\in\mathbb{N}} \sigma \left( \bigcup_{i=0}^n \pi_n^{-1}(\mathcal{A}) \right) .$$

Similarly, if we have a sequence of random variables $f_n:\Omega\to X$, the tail sigma-algebra on $\Omega$ is the one induced by the tail sigma-algebra on $X^\mathbb{N}$ via the map $(f_n):\Omega\to X^\mathbb{N}$.

An event (measurable subset) of the tail sigma-algebra is called a tail event.

(Similar definitions can be given for other cardinalities than $\mathbb{N}$.)

Properties

• Every shift-invariant event is a tail event.

• The converse is not true. For example, consider a sequence $(x_n)\in X^\mathbb{N}$ where the $x_n$ are distinct. The event of “having the same tail as $(x_n)$”, i.e. the set

  $$\{ (y_n) \,:\, \exists N, \forall n\ge N, y_n = x_n \} \;\subseteq\; X^\mathbb{N}$$

  is a tail event, but is not shift-invariant (if the $x_n$ are distinct).

• The Kolmogorov zero-one law says that for iid random variables, every tail event has probability zero or one.

See also

• probability theory, categorical probability
• zero-one law, exchangeability
• ergodicity, invariant measure, invariant set, ergodic decomposition theorem
• law of large numbers, ergodic theorem?

References

• Tobias Fritz and Eigil Fjeldgren Rischel, Infinite products and zero-one laws in categorical probability, Compositionality 2(3) 2020. (arXiv:1912.02769)

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

category: probability