We develop a strong approximation approach to extended multidimensional renewal theory. The consequences of this approximation are a Bahadur-Kiefer type representation of the renewal process in terms of partial sums, Strassen and Chung type laws of the iterated logarithm. We also give a characterization of the renewal process by four classes of deterministic curves in the sense of Revesz (1982). We generalize our results to the case of non-independent and/or nonidentically distributed random vectors.
"Strong Approximation of Extended Renewal Processes." Ann. Probab. 12 (4) 1149 - 1166, November, 1984. https://doi.org/10.1214/aop/1176993145