Tail sigma-algebra ($\mathcal{T}$)

For a sequence of random variables $(X_n)$, the tail sigma-algebra isIts events are unaffected by changing any finite initial segment.

For a sequence of finite real random variables, ${\mathrm{limsup}}_n{n}^{-1}{\sum }_{j=1}^n{X}_j$ is measurable with respect to its tail sigma-algebra: deleting finitely many summands changes each sample mean by a quantity tending to zero. When the random variables are independent random variables, the Kolmogorov zero-one law makes every finite such limit constant almost surely.

A tail event is measurable with respect to a tail sigma-algebra, so changing finitely many coordinates does not affect whether it occurs.

The tail sigma-algebra of a sequence of independent random variables is trivial: every tail event has probability zero or one.