A Renewal Theorem in the Infinite Mean Case

Abstract

Let $F(\cdot)$ be a c.d.f. on $(0, \infty)$ such that $1 - F(x)$ is regularly varying with exponent $-\alpha, \frac{1}{2} < \alpha \leq 1$. Let $Q(\cdot): \mathscr{R}^+ \rightarrow \mathscr{R}^+$ be nonincreasing and regularly varying with exponent $-\beta, 0 \leq \beta < 1$. Then, as $t \rightarrow \infty, (U \ast Q)(t) \equiv \int_{\lbrack 0,t\rbrack}Q(t - u)U(du)$ is asymptotic to $c(\alpha, \beta)(\int^t_0Q(u) du)(\int^t_0(1 - F(u)) du)^{-1}$, where $U(\cdot)$ is the renewal function associated with $F(\cdot)$ and $c(\alpha, \beta)$ is a suitable constant. This is an improved version of a theorem due to Teugels, whose proof appears to be incomplete. Applications of the result to the second order behavior of $U(t)$ in some special cases are also given.