Only the first term of some series counts

Abstract

Let X,X1,X2,... be i.i.d. random variables, and set Sn=X1+...+Xn. We prove that for three important distributions of X, namely normal, exponential and geometric, series of the type [summation operator]n>=1anP(Sn>=xbn) or [summation operator]n>=1anP(Sn>=xbn) behave like their first term as x-->[infinity].Tail probabilities of sums of i.i.d. random variables Normal distribution Exponential distribution Geometric distribution