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

Azmoodeh, E.; Gasbarra, D.

doi:10.1090/tpms/1062

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

Authors: E. Azmoodeh and D. Gasbarra
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
MathSciNet review: 3824678
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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 \begin{equation*} \mathscr {P}_\infty := \left \{P_n (x) = \mathfrak {R}^n_\infty \mathbf {1}\colon n \ge 1 \right \}. \end{equation*} 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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