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1 results

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

  • Part of
  • José A. Adell, Alberto Lekuona
  • Journal:
    Journal of Applied Probability / Volume 37 / Issue 3 / September 2000
    Published online by Cambridge University Press:
    14 July 2016, pp. 765-777
    Print publication:
    September 2000

  • In this paper, we consider positive linear operators L representable in terms 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 such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.