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Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

On a new Sheffer class of polynomials related to normal product distribution


Authors: E. Azmoodeh and D. Gasbarra
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 98 (2018).
Journal: Theor. Probability and Math. Statist. 98 (2019), 51-71
MSC (2010): Primary 60F05, 60G50, 46L54, 60H07, 26C10
DOI: https://doi.org/10.1090/tpms/1062
Published electronically: August 19, 2019
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, using the Stein operator $ \mathfrak{R}_\infty $ given in [Bernoulli 23 (2017), pp. 3311-3345], associated with the normal product distribution living in the second Wiener chaos, we introduce a new class of polynomials

$\displaystyle \mathscr {P}_\infty := \left \{P_n (x) = \mathfrak{R}^n_\infty \mathbf {1}\colon n \ge 1 \right \}.$    

We analyze in detail the polynomials class $ \mathscr {P}_\infty $, and relate it to Rota's Umbral calculus by showing that it is a Sheffer family and enjoys many interesting properties. Lastly, we study the connection between the polynomial class $ \mathscr {P}_\infty $ and the non-central probabilistic limit theorems within the second Wiener chaos.

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Additional Information

E. Azmoodeh
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, N-Süd UG/7, 44780 Bochum, Germany
Email: ehsan.azmoodeh@rub.de

D. Gasbarra
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin katu 2b) FI-00014 Helsinki, Finland
Email: dario.gasbarra@helsinki.fi

DOI: https://doi.org/10.1090/tpms/1062
Keywords: Second Wiener chaos, normal product distribution, cumulants/moments, weak convergence, Malliavin calculus, Sheffer polynomials, umbral calculus
Received by editor(s): February 19, 2018
Published electronically: August 19, 2019
Article copyright: © Copyright 2019 American Mathematical Society