On a renewal function when the second moment is infinite

Let {Sk}{Sk} be a random walk with independent, identically distributed real-valued increments {Xi}{Xi}, having a nonarithmetic distribution, finite expectation μ>0μ>0 and infinite moment Emax(0,X1)2Emax(0,X1)2. A refinement of the elementary renewal theorem is given in the following form: Emin{k:Sk>t}−t/μ∼ρ(t)as t→∞, where ρ(t)ρ(t) is a specific function such that ρ(t)→∞ρ(t)→∞ as t→∞t→∞.