Tail sigma-algebra
= Tail sigma-algebra
{title2=$\mathcal T$}
For a <sequence> of <random variables> $(X_n)_{n\geq1}$ on a <probability space> $(\Omega,\mathcal F,\mathbb P)$, define
$$
\mathcal G_n=\sigma(X_n,X_{n+1},\ldots),\qquad
\mathcal T=\bigcap_{n\geq1}\mathcal G_n.
$$
The decreasing <sigma-algebras> $\mathcal G_n$ are generated by the listed <random variables>. Their <intersection> consists of <events> expressible after discarding every finite prefix. No <independence> is required.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[Probability: Theory and Examples (fifth edition), §2.5 Convergence of Random Series, printed p. 81: definition of the tail sigma-field]{ref}
The point is to distinguish a long observation record from a property of an infinite <sequence>. The <event> of seeing infinitely many heads and the <event> that <sample means> converge survive any finite change, while an exact first-head time does not. <Kolmogorov's zero–one law> adds the striking conclusion that under <independence> each <tail event> has <probability> zero or one; the <tail sigma-algebra> itself can still contain many nonempty <sets>.