Theory of Probability and Mathematical Statistics
On a new Sheffer class of polynomials related to normal product distribution
E. Azmoodeh, D. Gasbarra
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Abstract: In this paper, using the Stein operator $\RR_\infty$ given in \cite{g-2normal}, associated with the normal product distribution living in the second Wiener chaos, we introduce a new class of polynomials \PP_\infty\coloneqq \left \{ P_n (x) = \RR^n_\infty \mathbf{1} \, : \, n \ge 1 \right \}. We analyze in details the polynomials class $\PP_\infty$, 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 $\PP_\infty$ and the non-central probabilistic limit theorems within the second Wiener chaos.
Keywords: Second Wiener chaos, normal product distribution, cumulants/moments, weak convergence, Malliavin calculus, Sheffer polynomials, umbral calculus.
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