Definition

tails

Let $\{X_n\}$ be a sequence of random variables and define $\mathcal{F}_{n}' = \sigma(X_{n+1}, X_{n+2}, \dots)$, $n = 1, 2, \dots$. The tail $\sigma$-field $\mathcal{T}$ is defined as $\mathcal{T} = \cap_{n}\mathcal{F}_{n}'$. If $A \in \mathcal{T}$ we call $A$ a tail event, and if a random variable is measurable with respect to $\mathcal{T}$ we call it a tail random variable.