Stability of one characterization by the properties of the renewal process (Q1925163)

Let \(\{S_n\), \(n\geq 1\); \(F(x)\}\) be a renewal process with continuous function \(F(x)\). For fixed \(t>0\) we call the random variables \(a(t)= t-S_{N(t)}\), \(b(t)= S_{N(t)+1} -t\), respectively, the age and the residual life at \(t\), where \(N(t)\) denotes the number of points of the process in the interval \((0,t)\). If \(a(t)\) and \(b(t)\) are almost independent in considered sense for all \(t\geq 0\), then there exists an absolute constant \(C\) such that \(\inf_{\lambda >0} \sup_{x \geq 0} |1- F(x)- \exp (-\lambda x) |\leq C\varepsilon\).