Functional approximation theorems for controlled renewal processes

We prove a functional law of large numbers and a functional central limit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith interarrival time is scaled by a function of the number of previous i arrivals. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solution of an ordinary differential equation. Likewise, the functional central limit theorem establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation. Our proofs are based on martingale and time-change arguments.