A limit theorem on subintervals of interrenewal times

Consider a renewal process {X_n, n ≥ 1} for which there is defined an associated sequence of independent and identically distributed random variables {B_n, n ≥ 1} such that B_n is the length of a subinterval of X_n. We show that when attention is restricted only to B-intervals, the asymptotic joint distribution of the residual life and total life of a B-interval is that of a renewal process generated by {B_n, n ≥ 1}.