Stone has decomposed the renewal measure of a probability distribution $F$ into two parts: a finite component with the same tail behaviour as that of $F$ and an absolutely continuous one which is nearly stationary at infinity. Our theorem asserts the uniformity of this decomposition.
"Uniformity in Stone's Decomposition of the Renewal Measure." Ann. Probab. 5 (4) 560 - 564, August, 1977. https://doi.org/10.1214/aop/1176995761