Definition:Tail Sigma-Algebra

Definition

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\sequence {X_n}_{n \mathop \in \N}$ be a sequence of real-valued random variables on $\struct {X, \Sigma}$.

Let:

$\TT_n = \map \sigma {\set {X_k: k \in \N, \, k \ge n + 1} }$

where $\map \sigma {\set {X_k: k \in \N, \, k \ge n + 1} }$ denotes the $\sigma$-algebra generated by $\set {X_k : k \in \N, \, k \ge n + 1}$.

Define:

$\ds \TT = \bigcap_{n \mathop = 1}^\infty \TT_n$

Then we say that $\TT$ is the tail $\sigma$-algebra of $\sequence {X_n}_{n \mathop \in \N}$.

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $4.10$: Tail $\sigma$-algebras