A remark on the tail probability of a distribution

AbstractLet {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ϵ, let Sn = Σj=1n Xj and E{N∞(r, t, ϵ)} = Σn=1∞ nr−2P{|Sn| > ϵnrt}. In this paper, we prove that (1) limϵ→0+ ϵα(r−1)E{N∞(r, t, ϵ)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, K(r, t) = {2α(r−1)2Γ((1 + α(r − 1))2)}{(r − 1) Γ(12)}, and α = 2t(2r − t); (2) limϵ→0+ G(t, ϵ)H(t, ϵ) = 0 if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ϵ) = E{N∞(t, t, ϵ)} = Σn=1∞ nt−2P{| Sn | > ϵn} → ∞ as ϵ → 0+ and H(t, ϵ) = E{N∞(t, t, ϵ)} = Σn=1∞ nt−2P{| Sn | > ϵn2t} → ∞ as ϵ → 0+, i.e., H(t, ϵ) goes to infinity much faster than G(t, ϵ) as ϵ → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution