= Tail event
{wiki}
A tail event is an <event> $A\in\mathcal T$, where $\mathcal T$ is the <tail sigma-algebra> of the chosen <sequence> of <random variables>. It concerns information available arbitrarily far along the <sequence>. In the measurable coordinate models described in <finite-prefix invariance of tail events>, any finite edit preserves whether it occurs.
This is stronger than merely needing a long waiting time. On $\{0,1\}^{\mathbb N}$, with $1$ denoting heads, let $K=\inf\{j\geq1:X_j=1\}$, taking $K=\infty$ if there is no head. The <event>
$$
\{K=3\}=\{X_1=0,X_2=0,X_3=1\}
$$
is not a <tail event>: changing only $X_1$ from zero to one changes membership. Under fair <independent> <Bernoulli trials> its <probability> is $1/8$. Even the <event> $\{K<\infty\}$, of <probability> one in this model, is not a <tail event> as a literal <set>: the all-zero <sequence> and the <sequence> with just one initial head differ in one coordinate. A <probability> of zero or one does not characterize <tail events>.
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