Inverspositive Operatoren und Taubersätze in der Erneuerungstheorie

Abstract

A generalization of ordinary renewal processes has been treated byHinderer and the author: for the stochastic process with values in ℝ₊ the aim is (a,a+β) with 0≦a<∞, 0<β≦∞ instead of (a, ∞), and a possible passing over the aim is replaced by a phase of stagnation. In the present paper for the open case of an infinite mean length of the undisturbed steps the relationV(a)/a→0(a→∞), with β fixed, for the mean waiting timeV(a) until reaching the aim is obtained. The proof uses the theory of inversepositive operators ofCollatz andSchröder. This concept also yields an elementary proof of the so-called elementary renewal theorem. Finally the Tauberian theorem ofIkehara inAgmon's version is generalized; it yields the so-called general renewal theorem in the case of finite mean length of the steps, which is also a corollary of a general Tauberian theorem ofBeneš.