On a renewal function when the second moment is infinite

Abstract

Let {Sk} be a random walk with independent, identically distributed real-valued increments {Xi}, having a nonarithmetic distribution, finite expectation [mu]>0 and infinite moment Emax(0,X1)2. A refinement of the elementary renewal theorem is given in the following form: where [rho](t) is a specific function such that [rho](t)-->[infinity] as t-->[infinity].