Some results on the joint distribution of the renewal Epochs prior to a given time instant

Let (Sn)n[greater-or-equal, slanted]0 be a renewal process with interarrival times X1,X2,... Several results on the behavior of the renewal process up to a given time t>0 or up to a given Sn=s are proved. For example, X1 is stochastically dominated by XN(t)+1, and X0=0, X1,...,XN(t)+1 is a stochastically increasing sequence, where N(t)=sup{n[greater-or-equal, slanted]0|Sn[less-than-or-equals, slant]t}. Conditions are given under which the distribution of the process (S[nt])0[less-than-or-equals, slant]t[less-than-or-equals, slant]1, given that Sn=s, converges weakly in D[0,1] to the point mass at the function xs(t)=st. The result e.g. holds, if X1 has a strongly unimodal distribution or if E(X21|S2)[less-than-or-equals, slant]S22/(2(1+c)) a.s. for some c>0. In this context some new characterizations of the gamma, Poisson, binomial and negative binomial distributions are derived.