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:


 * ${\mathcal T}_n = \map \sigma {\set {X_k : k \in \N, \, k \ge n + 1} }$

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:


 * $\ds \mathcal T = \bigcap_{n \mathop = 1}^\infty {\mathcal T}_n$

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