A Sufficient Condition for Association of a Renewal Process

Abstract

Let $\{N(t): t \geq 0\}$ be a renewal counting process with lifetime density $f(t)$. For each bounded Borel set $A$ contained in $\lbrack 0, \infty)$, denote the number of renewals in $A$ by $N(A)$. The renewal process is called associated if the corresponding family of random variables, $N(A)$, is associated. The result of this note is that the renewal process is associated whenever $\log(f)$ is a convex function (which implies a decreasing failure rate).