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Let γ t and δ t denote the residual life at t and current life at t, respectively, of a renewal process 
, with 
the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as 
holds for a single positive integer n, then 
is a Poisson process. On the other hand, for a delayed renewal process 
with 
the residual life at t, we find that for some fixed positive integer n, if 
is independent of t, then 
is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t and δ t.
