Renewal theory for random variables with a heavy tailed distribution and finite variance

Let X1,X2,…XnX1,X2,…Xn be independent and identically distributed (i.i.d.) non-negative random variables with a common distribution function (d.f.) FF with unbounded support and EX12<∞. We show that for a large class of heavy tailed random variables with a finite variance the renewal function UU satisfies U(x)−xμ−μ22μ2∼−1μx∫x∞∫s∞(1−F(u))duds as x→∞x→∞.