Taylor's formula and preservation of generalized convexity for positive linear operators

In this paper, we consider positive linear operators L representable in ter ms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for su ch operators, thus extending to a positive linear operator setting the clas sical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generali zed convexity of order n. We illustrate the preceding results by considerin g discrete time processes, counting and renewal processes, centred subordin ators and the Yule birth process.