On a weighted embedding for generalized pontograms

A weighted embedding for the generalized pontogram {Kn(t): 0⩽t⩽1} corresponding pointwise to a renewal process {N(s): 0⩽s<∞} via Kn(t)=n−1/2(N(nt)−tN(n)) is studied in this paper. After proper normalization, weak convergence results for the processes {Kn(t): 0⩽t⩽1} are derived both in sup-norm as well as in Lp-norm. These results are suggested to serve as asymptotic testing devices for detecting changes in the intensity of the underlying renewal process.