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

Part of: Distribution theory - Probability Approximations and expansions

Published online by Cambridge University Press:  14 July 2016

José A. Adell  and
Alberto Lekuona
José A. Adell*
Affiliation:
Universidad de Zaragoza
Alberto Lekuona*
Affiliation:
Universidad de Zaragoza
*
∗Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
∗Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
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Abstract

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.

Keywords

Taylor's formulapositive linear operatorpreservation propertiesstochastic orderderived processsubordinator

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 41A36: Approximation by positive operators
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 765 - 777
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

Research supported by the DGICYT PB98-1577-C02-01 Grant.

References

Abate, A., and Whitt, W. (1996). An operational calculus for probability distributions via Laplace transforms. Adv. Appl. Prob. 28, 75–113.Google Scholar
Adell, J. A., and Perez-Palomares, A. (1999). Stochastic orders in preservation properties by Bernstein-type operators. Adv. Appl. Prob. 31, 492–509.CrossRefGoogle Scholar
Altomare, F., and Campiti, M. (1994). Korovkin-type Approximation Theory and its Applications. Walter de Gruyter, Berlin.CrossRefGoogle Scholar
De la Cal, J., and Carcamo, J. (1999). Preservation of p-continuity by Bernstein-type operators. J. Approx. Theory 101, 156–159.Google Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Makowski, A. M., and Philips, T. K. (1992). Stochastic convexity of sums of i.i.d. non-negative random variables with applications. J. Appl. Prob. 29, 156–167.Google Scholar
Nanda, A. K., Jain, K., and Singh, H. (1996). On closure of some partial orderings under mixtures. J. Appl. Prob. 33, 698–706.Google Scholar
Roberts, A. W., and Varberg, D. E. (1973). Convex Functions. Academic Press, New York.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1988). Stochastic convexity and its applications. Adv. Appl. Prob. 20, 427–446.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Ordering and its Applications. Academic Press, Boston.Google Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes in Statist. 97). Springer, Berlin.Google Scholar