On a weighted embedding for pontograms

A generalized pontogram {Kn(t): 0 [less-than-or-equals, slant] t [less-than-or-equals, slant] 1} corresponding pointwise to a renewal counting process {N(x): 0 [less-than-or-equals, slant] x < [infinity]} via Kn(t) = n-1/2(N(nt)-tN(n)) is investigated in this paper. A weighted embedding for the process {Kn(t): 0 [less-than-or-equals, slant] t [less-than-or-equals, slant] 1} is studied. After proper normalization, weak convergence results for the processes {Kn(t): 0 [less-than-or-equals, slant] t [less-than-or-equals, slant] 1} are derived both in sup-norm as well as in Lp-norm.