Invariance principles for renewal processes when only moments of low order exist (Q1116529)

Let \((U_ i)_{i=1,2,...}\) be a sequence of i.i.d. r.v.'s with E \(U_ i>0\), \(0<D u_ i<\infty\) and \(R_ n=U_ 1+...+U_ n\), \(n=1,2,...\), \(R_ 0=0\). Let \((N(t))_{t\geq 0}\) be a renewal process based on the sequence \((U_ i)_{i=1,2,...}\), i.e., \[ N(t)=\min \{n:R_ n>t\},\quad t\geq 0. \] The aim of this paper is to prove strong invariance principles for \((N(t))_{t\geq 0}\) under minimal conditions for the moments of \((U_ i)_{i=1,2,...}\) and to get some weak invariance principles, i.e. approximations in terms of convergence in probability. As an application, a Darling-Erdős [\textit{D. A. Darling} and \textit{P. Erdős}, Duke Math. J. 23, 143-155 (1956; Zbl 0070.138)] type theorem for renewal processes is given.