Tail Sigma Algebra

Definition

The tail sigma algebra of a sequence of random variables is the sigma algebra which is the intersection of the sigma algebras of all final parts of the sequence. A tail event is an element of the tail sigma-algebra.

The tail sigma algebra coincides with the sigma algebra generated by the union of the sigma algebras of each of of the random variables.

Notation

Let $\seq{f}$ be a sequence of random variables. Denote the tail sigma algebra by $T(\seq{f})$. We defined it as:

\[ T(\seq{f}) = \bigcap_{n = 1}^{\infty} \sigma (\set{X_{n+k}}_k). \]

Results