We present a general methodology for proving invariance principles for renewal processes, resulting in almost sure and probability inequality approximations. We show that our obtained rates are best possible in the i.i.d. case.
"Invariance Principles for Renewal Processes." Ann. Probab. 15 (4) 1441 - 1460, October, 1987. https://doi.org/10.1214/aop/1176991986