Strong approximation of renewal processes

AbstractWe develop a strong approximation of renewal processes. The consequences of this approximation are laws of the iterated logarithm and a Bahadur–Kiefer representation ofthe renewal process in terms of partial sums. The Bahadur–Kiefer representation implies that the rate of the strong approximation with the same Wiener process for both partial sums and renewal processes cannot be improved upon when the underlying random variables have finite fourth moments. We can generalize our results to the case of nonindependent and/or nonidentically distributed random variables