On a renewal function when the second moment is infinite

Abstract

Let { S k } be a random walk with independent, identically distributed real-valued increments { X i } , having a nonarithmetic distribution, finite expectation μ > 0 and infinite moment E max ( 0 , X 1 ) 2 . A refinement of the elementary renewal theorem is given in the following form: E min { k : S k > t } − t / μ ∼ ρ ( t ) as t → ∞ , where ρ ( t ) is a specific function such that ρ ( t ) → ∞ as t → ∞ .