Intuition behind tail events? I am studying now Probability Theory and when I came to Kolmogorov's law 0-1 I met with tail events. Despite, formal definition it is little hard to understand the intuition of tail events. I am asking a favour for those who are more deep in probability theory to explain tail events and may this question can be useful also for the others. Thanks in advance.
 A: Let $(X_1,X_2,\ldots)$ be a sequence of random variables defined on some probability space $(\Omega,\mathcal F,\Bbb P)$. You can think of an event $A\in\sigma(X_1,X_2,\ldots)$ as a property $p_A$ that the sequence of numbers $\Bbb X(\omega):=(X_1(\omega),X_2(\omega),\ldots)$ may or may not enjoy: $\omega\in A$ if and if $\Bbb X(\omega)$ has property $p_A$. Some properties depend on the entirety of the sequence:
$$
p_{1}\hbox{ is the property that }\sum_{n=1}^\infty X_n(\omega)\le 17.
$$
A change in even one $X_n(\omega)$ may affect whether $\Bbb X(\omega)$ has property $p_1$.
Other properties depend only on the asymptotic behavior of $\Bbb X(\omega)$:
$$
p_{2}\hbox{ is the property that }\sum_{n=1}^\infty X_n(\omega)<\infty
$$
is one such property. A change in the first few terms of $\Bbb X(\omega)$ does not affect whether the sum is finite or not. Properties of this asymptotic type correspond to  tail events.
Tail events are   saturated for "asymptotic equivalence" in the sense that if $B$ is a tail event and $\omega\in B$, and if $\omega'\in\Omega$ is another sample point such that $X_k(\omega')=X_k(\omega)$ for all $k\ge K(\omega,\omega')$ then $\omega'\in B$ as well. 
