For an urn model (arising typically in the sequential estimation of the size of a finite population), along with an invariance principle for a partial sequence of nonnegative random variables, a renewal theorem relating to some stopping times is established. A representation of these random variables in terms of linear combinations of some martingale-differences provides the key to a simple solution.
"A Renewal Theorem for an Urn Model." Ann. Probab. 10 (3) 838 - 843, August, 1982. https://doi.org/10.1214/aop/1176993794