A remark on the tail probability of a distribution

Let {Xn}n>=1 be a sequence of independent and identically distributed random variables. For each integer n >= 1 and positive constants r, t, and [epsilon], let Sn = [Sigma]j=1n Xj and E{N[infinity](r, t, [epsilon])} = [Sigma]n=1[infinity] nr-2P{Sn > [epsilon]nr/t}. In this paper, we prove that (1) lim[epsilon]-->0+ [epsilon][alpha](r-1)E{N[infinity](r, t, [epsilon])} =K(r, t) if E(X1) = 0, Var(X1) = 1, and E( X1 t) < [infinity], where 2 <= t < 2r <= 2t, , and [alpha] = 2t/(2r - t); (2) lim[epsilon]-->0+ G(t, [epsilon])/H(t, [epsilon]) = 0 if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(X1t) < [infinity], where G(t, [epsilon]) = E{N[infinity](t, t, [epsilon])} = [Sigma]n=1[infinity] nt-2P{ Sn > [epsilon]n} --> [infinity] as [epsilon] --> 0+ and H(t, [epsilon]) = E{N[infinity](t, t, [epsilon])} = [Sigma]n=1[infinity] nt-2P{ Sn > [epsilon]n2/t} --> [infinity] as [epsilon] --> 0+, i.e., H(t, [epsilon]) goes to infinity much faster than G(t, [epsilon]) as [epsilon] --> 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E( X1 t) < [infinity]. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.