By using a symbolic technique known in the literature as the classical umbral calculus, we characterize two classes of polynomials related to Lévy processes: the Kailath-Segall and the time-space harmonic polynomials. We provide the Kailath-Segall formula in terms of cumulants and we recover simple closed-forms for several families of polynomials with respect to not centered Lévy processes, such as the Hermite polynomials with Brownian motion, Poisson-Charlier polynomials with Poisson processes, actuarial polynomials with Gamma processes, first kind Meixner polynomials with Pascal processes, and Bernoulli, Euler, and Krawtchuk polynomials with suitable random walks. © 2013 Copyright Taylor and Francis Group, LLC.
On some applications of a symbolic representation of non centered Lévy processes / Elvira Di, Nardo; Oliva, I.. - In: COMMUNICATIONS IN STATISTICS. THEORY AND METHODS. - ISSN 0361-0926. - 42:21(2013), pp. 3974-3988. [10.1080/03610926.2011.642920]
On some applications of a symbolic representation of non centered Lévy processes
Oliva I.
2013
Abstract
By using a symbolic technique known in the literature as the classical umbral calculus, we characterize two classes of polynomials related to Lévy processes: the Kailath-Segall and the time-space harmonic polynomials. We provide the Kailath-Segall formula in terms of cumulants and we recover simple closed-forms for several families of polynomials with respect to not centered Lévy processes, such as the Hermite polynomials with Brownian motion, Poisson-Charlier polynomials with Poisson processes, actuarial polynomials with Gamma processes, first kind Meixner polynomials with Pascal processes, and Bernoulli, Euler, and Krawtchuk polynomials with suitable random walks. © 2013 Copyright Taylor and Francis Group, LLC.File | Dimensione | Formato | |
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