Given a sequence of random variables X_{1},X_{2},..., call an event "in F_{k}" if it can be expressed as a list of conditions on {X_{1},...,X_{k}}. 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 F_{80000}, 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}^{∞}F_{k} (i.e. it contains all events that are in some F_{k} 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 F_{k} for all k; equivalently, E is independent of all the X's.

As an example, let each X_{j} 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 X_{n}, lim inf X_{n}, lim (X_{1}+...+X_{n})/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.