Some results on the joint distribution of the renewal epochs prior to a given time instant
Let \((S_ n)_{n\geq 0}\) be a renewal process with interarrival times \(X_ 1,X_ 2,...\). Several results on the behavior of the renewal process up to a given time \(t>0\) or up to a given \(S_ n=s\) are proved. For example, \(X_ 1\) is stochastically dominated by \(X_{N(t)+1}\), and \(X_ 0=0,X_ 1,...,X_{N(t)+1}\) is a stochastically increasing sequence, where \(N(t)=\sup \{n\geq 0| S_ n\leq t\}\). Conditions are given under which the distribution of the process \((S_{[nt]})_{0\leq t\leq 1}\), given that \(S_ n=s\), converges weakly in D[0,1] to the point mass at the function \(x_ s(t)=st\). The result e.g. holds, if \(X_ 1\) has a strongly unimodal distribution or if \(E(X^ 2_ 1| S_ 2)\leq S^ 2_ 2/(2(1+c))\) a.s. for some \(c>0\). In this context some new characterizations of the gamma, Poisson, binomial and negative binomial distributions are derived.
binomial and negative binomial distributions
scientific article; zbMATH DE number 4064212