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
Full-text PDF
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 \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.
References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- B. Arras, E. Azmoodeh, G. Poly, and Y. Swan, Stein characterizations for linear combinations of gamma random variables, 2017, https://arxiv.org/pdf/1709.01161.pdf.
- Benjamin Arras, Ehsan Azmoodeh, Guillaume Poly, and Yvik Swan, A bound on the Wasserstein-2 distance between linear combinations of independent random variables, Stochastic Process. Appl. 129 (2019), no. 7, 2341–2375. MR 3958435, DOI https://doi.org/10.1016/j.spa.2018.07.009
- Florin Avram and Murad S. Taqqu, Noncentral limit theorems and Appell polynomials, Ann. Probab. 15 (1987), no. 2, 767–775. MR 885142
- E. Azmoodeh and D. Gasbarra, New moments criteria for convergence towards normal product/tetilla laws, 2017, https://arxiv.org/abs/1708.07681.
- Ehsan Azmoodeh, Dominique Malicet, Guillaume Mijoule, and Guillaume Poly, Generalization of the Nualart-Peccati criterion, Ann. Probab. 44 (2016), no. 2, 924–954. MR 3474463, DOI https://doi.org/10.1214/14-AOP992
- Ehsan Azmoodeh, Giovanni Peccati, and Guillaume Poly, Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach, In memoriam Marc Yor—Séminaire de Probabilités XLVII, Lecture Notes in Math., vol. 2137, Springer, Cham, 2015, pp. 339–367. MR 3444306, DOI https://doi.org/10.1007/978-3-319-18585-9_16
- Shuyang Bai and Murad S. Taqqu, Behavior of the generalized Rosenblatt process at extreme critical exponent values, Ann. Probab. 45 (2017), no. 2, 1278–1324. MR 3630299, DOI https://doi.org/10.1214/15-AOP1087
- V. Bally, Introduction to Malliavin Calculus, 2007, http://perso-math.univ-mlv.fr/users/bally.vlad/Osaka100407.pdf.
- Jan Baldeaux and Eckhard Platen, Functionals of multidimensional diffusions with applications to finance, Bocconi & Springer Series, vol. 5, Springer, Cham; Bocconi University Press, Milan, 2013. MR 3113191
- Árpád Baricz and Saminathan Ponnusamy, On Turán type inequalities for modified Bessel functions, Proc. Amer. Math. Soc. 141 (2013), no. 2, 523–532. MR 2996956, DOI https://doi.org/10.1090/S0002-9939-2012-11325-5
- Youssèf Ben Cheikh, Some results on quasi-monomiality, Appl. Math. Comput. 141 (2003), no. 1, 63–76. Advanced special functions and related topics in differential equations (Melfi, 2001). MR 1984228, DOI https://doi.org/10.1016/S0096-3003%2802%2900321-1
- Youssèf Ben Cheikh and Khalfa Douak, On two-orthogonal polynomials related to the Bateman’s $J_n^{u,v}$-function, Methods Appl. Anal. 7 (2000), no. 4, 641–662. MR 1868550, DOI https://doi.org/10.4310/MAA.2000.v7.n4.a3
- T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR 0481884
- Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128
- Aurélien Deya and Ivan Nourdin, Convergence of Wigner integrals to the tetilla law, ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012), 101–127. MR 2893412
- Robert E. Gaunt, On Stein’s method for products of normal random variables and zero bias couplings, Bernoulli 23 (2017), no. 4B, 3311–3345. MR 3654808, DOI https://doi.org/10.3150/16-BEJ848
- Robert E. Gaunt, Variance-gamma approximation via Stein’s method, Electron. J. Probab. 19 (2014), no. 38, 33. MR 3194737, DOI https://doi.org/10.1214/EJP.v19-3020
- Peter Eichelsbacher and Christoph Thäle, Malliavin-Stein method for variance-gamma approximation on Wiener space, Electron. J. Probab. 20 (2015), Paper No. 123, 28. MR 3425543, DOI https://doi.org/10.1214/EJP.v20-4136
- Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey; Reprint of the 2005 original. MR 2542683
- H. van Haeringen and L. P. Kok, Table errata: Table of integrals, series, and products [corrected and enlarged edition, Academic Press, New York, 1980; MR 81g:33001] by I. S. Gradshteyn [I. S. Gradshteĭn] and I. M. Ryzhik, Math. Comp. 39 (1982), no. 160, 747–757. MR 669666, DOI https://doi.org/10.1090/S0025-5718-1982-0669666-2
- Paul Malliavin, Integration and probability, Graduate Texts in Mathematics, vol. 157, Springer-Verlag, New York, 1995. With the collaboration of Hélène Airault, Leslie Kay and Gérard Letac; Edited and translated from the French by Kay; With a foreword by Mark Pinsky. MR 1335234
- Ivan Nourdin and Giovanni Peccati, Normal approximations with Malliavin calculus, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012. From Stein’s method to universality. MR 2962301
- Ivan Nourdin and Guillaume Poly, Convergence in law in the second Wiener/Wigner chaos, Electron. Commun. Probab. 17 (2012), no. 36, 12. MR 2970700, DOI https://doi.org/10.1214/ecp.v17-2023
- David Nualart and Giovanni Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005), no. 1, 177–193. MR 2118863, DOI https://doi.org/10.1214/009117904000000621
- Giovanni Peccati and Murad S. Taqqu, Wiener chaos: moments, cumulants and diagrams, Bocconi & Springer Series, vol. 1, Springer, Milan; Bocconi University Press, Milan, 2011. A survey with computer implementation; Supplementary material available online. MR 2791919
- Steven Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. MR 741185
- Steven Roman, The theory of the umbral calculus. I, J. Math. Anal. Appl. 87 (1982), no. 1, 58–115. MR 653607, DOI https://doi.org/10.1016/0022-247X%2882%2990154-8
- Steven M. Roman and Gian-Carlo Rota, The umbral calculus, Advances in Math. 27 (1978), no. 2, 95–188. MR 485417, DOI https://doi.org/10.1016/0001-8708%2878%2990087-7
- Robert J. Serfling, Approximation theorems of mathematical statistics, John Wiley & Sons, Inc., New York, 1980. Wiley Series in Probability and Mathematical Statistics. MR 595165
- I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 0000081
- M. D. Springer and W. E. Thompson, The distribution of products of beta, gamma and Gaussian random variables, SIAM J. Appl. Math. 18 (1970), 721–737. MR 264733, DOI https://doi.org/10.1137/0118065
- W. Van Assche and S. B. Yakubovich, Multiple orthogonal polynomials associated with Macdonald functions, Integral Transform. Spec. Funct. 9 (2000), no. 3, 229–244. MR 1782974, DOI https://doi.org/10.1080/10652460008819257
References
- M. Abramowitz and I. A. Stegun, eds., Modified Bessel Functions I and K, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed., Dover, New York, 1972, pp. 374–377. MR 1225604
- B. Arras, E. Azmoodeh, G. Poly, and Y. Swan, Stein characterizations for linear combinations of gamma random variables, 2017, https://arxiv.org/pdf/1709.01161.pdf.
- B. Arras, E. Azmoodeh, G. Poly, and Y. Swan, A bound on the 2-Wasserstein distance between linear combinations of independent random variables, Stochastic Process. Appl. 129 (2019), no. 7, 2341–2375. MR 3958435
- F. Avram and M. Taqqu, Noncentral limit theorems and Appell polynomials, Ann. Probab. 15 (1987), no. 2, 767–775. MR 885142
- E. Azmoodeh and D. Gasbarra, New moments criteria for convergence towards normal product/tetilla laws, 2017, https://arxiv.org/abs/1708.07681.
- E. Azmoodeh, D. Malicet, G. Mijoule, and G. Poly, Generalization of the Nualart–Peccati criterion, Ann. Probab. 44 (2016), no. 2, 924–954. MR 3474463
- E. Azmoodeh, G. Peccati, and G. Poly, Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach, Séminaire de Probabilités XLVII (Special volume in memory of Marc Yor), 2014, 339–367. MR 3444306
- S. Bai and M. Taqqu, Behavior of the generalized Rosenblatt process at extreme critical exponent values, Ann. Probab. 45 (2017), no. 2, 1278–1324. MR 3630299
- V. Bally, Introduction to Malliavin Calculus, 2007, http://perso-math.univ-mlv.fr/users/bally.vlad/Osaka100407.pdf.
- J. Baldeaux and E. Platen, Functionals of Multidimensional Diffusions with Applications to Finance, Bocconi & Springer Series, vol. 5, Springer & Bocconi University Press, 2013. MR 3113191
- Á. Baricz and S. Ponnusamy, On Turán type inequalities for modified Bessel functions, Proc. Amer. Math. Soc. 141 (2013), no. 2, 523–532. MR 2996956
- Y. Ben Cheikh, Some results on quasi-monomiality, Appl. Math. Comput. 141 (2003), no. 1, 63–76. MR 1984228
- Y. Ben Cheikh and K. Douak, On two-orthogonal polynomials related to the Bateman’s $J^{u,v}_n$-function, Methods Appl. Anal. 7 (2009), no. 4, 641–662. MR 1868550
- T. S. Chihara, An Introduction to Orthogonal Polynomials, Dover Books on Mathematics, 2011. MR 0481884
- L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, D. Reidel Publishing Co., 1974. MR 0460128
- A. Deya and I. Nourdin, Convergence of Wigner integrals to the tetilla law, ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012), 101–127. MR 2893412
- R. E. Gaunt, On Stein’s method for products of normal random variables and zero bias couplings, Bernoulli 23 (2017), no. 4B, 3311–3345. MR 3654808
- R. E. Gaunt, Variance-Gamma approximation via Stein’s method, Electron. J. Probab. 19 (2014), no. 38, 1–33. MR 3194737
- P. Eichelsbacher and C. Thäle, Malliavin–Stein method for Variance-gamma approximation on Wiener space, Electron. J. Probab. 20 (2015), no. 123, 1–28. MR 3425543
- M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, 2009. MR 2542683
- S. Gradshetyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed., Academic Press, 2007. MR 669666
- P. Malliavin, Integration and Probability, Graduate Texts in Mathematics, vol. 157, Springer-Verlag, New York, 1995. MR 1335234
- I. Nourdin and G. Peccati, Normal Approximations Using Malliavin Calculus: from Stein’s Method to Universality, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, 2012. MR 2962301
- I. Nourdin and G. Poly, Convergence in law in the second Wiener/Wigner chaos, Elect. Comm. in Probab. 17 (2012), no. 36. MR 2970700
- D. Nualart and G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 330 (2005), no. 1, 177–193. MR 2118863
- G. Peccati and M. S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams, Bocconi & Springer Series, vol. 1, Springer, Milan, 2011. MR 2791919
- S. Roman, The Umbral Calculus, Pure and Applied Mathematics, vol. 111, Academic Press, New York, 1984. MR 741185
- S. Roman, The theory of the umbral calculus, I. J. Math. Anal. 87 (1982), no. 1, 58–115. MR 653607
- S. Roman and G. Rota, The Umbral Calculus, Advances Math. 27 (1978), 95–188. MR 0485417
- R. J. Serfling, Approximation Theorems of Mathematical Statistics, John Wiley & Sons, 1980. MR 595165
- I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 0000081
- M. D. Springer and W. E. Thompson, The distribution of products of Beta, Gamma and Gaussian random variables, SIAM J. Appl. Math. 18 (1970), 721–737. MR 0264733
- W. Van Assche and S. B. Yakubovich, Multiple orthogonal polynomials associated with Macdonald functions, Integral Transforms Special Funct. 9 (2000), 229–244. MR 1782974
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60F05,
60G50,
46L54,
60H07,
26C10
Retrieve articles in all journals
with MSC (2010):
60F05,
60G50,
46L54,
60H07,
26C10
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
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