Given a sequence of random variables X1,X2,..., call an event "in Fk" if it can be expressed as a list of conditions on {X1,...,Xk}. For instance, if the X's are IID fair coin tosses, then the event "the first 80000 X's are bits encoding the first 10000 ASCII characters of Hamlet" is in F80000, as is the event "between 35000 and 45000 of the first 70000 X's were heads". The sequence generates the σ algebra F that is the smallest σ algebra containing all events in ∪k=1∞Fk (i.e. it contains all events that are in some Fk for some k). An event E is called a tail event of the sequence of X's if E is independent of all events in Fk for all k; equivalently, E is independent of all the X's.
As an example, let each Xj be a random real number chosen according to some distribution (they need not be independently chosen, or even follow the same distribution; just pick them as you like). Then the event that any of the limits lim Xn, lim inf Xn, lim (X1+...+Xn)/n (and many others) exists does not depend on the values of any of the X's, so it's a tail event.
For the
probability theorists among the readers, here's a slightly more comprehensive definition. Given a
sequence of
increasing σ algebras F
1⊂F
2⊂..., let F be the σ algebra generated by the F
k's ({F
k}
k is a
filtration of F). An event E is a
tail event of F if E is independent of F
k for all k.
Kolmogorov's 0-1 law applies to tail events, hence their importance.