Brown (1980), (1981) proved that the renewal function is concave if the interrenewal distribution is DFR (decreasing failure rate), and conjectured the converse. This note settles Brown's conjecture with a class of counterexamples. We also give a short proof of Shanthikumar's (1988) result that the DFR property is closed under geometric compounding.
"Concave renewal functions do not imply DFR interrenewal times." J. Appl. Probab. 48 (2) 583 - 588, June 2011. https://doi.org/10.1239/jap/1308662647