On the rate of convergence of the Ross approximation to the renewal function

Consider a renewal process {N(t), t greater than or equal to 0}. For fixed t > 0 and each n greater than or equal to 1, let Y-n,Y-1,..., Y-n,Y-n be independent exponentials each having mean t/n, independent of the renewal process. Ross [2] developed a recursion for the sequence of approximations m(n) = EN(Y-n,Y-1 + ... Y-n,Y-n) that converges to m(t) if the renewal function m(.) = EN(.) is continuous at t > 0. In this note, we derive an upper bound on the rate of convergence of this sequence under mild conditions on m near t. Tightness of this bound is discussed in terms of regularity conditions on m.