FUNCTIONAL APPROXIMATION THEOREMS FOR CONTROLLED RENEWAL PROCESSES

We prove a functional law of large numbers and a functional central li mit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith intera rrival time is scaled by a function of the number of previous i arriva ls. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solut ion of an ordinary differential equation. Likewise, the functional cen tral limit theorem establishes that the error in the law of large numb ers converges weakly to the solution of a stochastic differential equa tion. Our proofs are based on martingale and time-change arguments.