Discretisations of higher order and the theorems of Faà di Bruno and DeMoivre-Laplace

Imme van den Berg


We study discrete functions on equidistant and non-equidistant infinitesimal grids. We consider its difference quotients of higher order and give conditions for their near-equality to the corresponding derivatives. Important tools are the formula of Faà di Bruno for higher order derivatives and a discrete version of it. As an application of such transitions from the discrete to the continuous we extend the DeMoivre-Laplace Theorem to higher order: n-th order difference quotients of the binomial probability distribution tend to the corresponding n-th order partial differential quotients of the Gaussian distribution.

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DOI: https://doi.org/10.4115/jla.2013.5.6

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Journal of Logic and Analysis ISSN:  1759-9008