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Convergence of the fourth moment and infinite divisibility

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EN
Abstrakty
EN
In this note we prove that, for infinitely divisible laws, convergence of the fourth moment to 3 is sufficient to ensure convergence in law to the Gaussian distribution. Our results include infinitely divisible measures with respect to classical, free, Boolean and monotone convolution. A similar criterion is proved for compound Poissons with jump distribution supported on a finite number of atoms. In particular, this generalizes recent results of Nourdin and Poly (2012).
Rocznik
Strony
201--212
Opis fizyczny
Bibliogr. 24 poz.
Twórcy
autor
  • Department of Mathematics, Saarland University, Saarbrücken, Germany
Bibliografia
  • [1] O. Arizmendi, T. Hasebe, and N. Sakuma, On the law of free subordinators, ALEA Lat. Am. J. Probab. Math. Stat. 10 (1) (2013), pp. 271-291.
  • [2] Z. D. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, second edition, Springer, New York 2009.
  • [3] O. E. Barndorff-Nielsen and S. Thorbjørnsen, Classical and free infinite divisibility and Lévy processes, in: Quantum Independent Increment Processes II. Quantum Lévy Processes, Classical Probability and Applications to Physics, U. Franz and M. Schürmann (Eds.), Springer, 2006, pp. 33-160.
  • [4] S. T. Belinschi, Complex analysis methods in noncommutative probability, available in arXiv:math/0602343v1.
  • [5] H. Bercovici and V. Pata, Stable laws and domains of attraction in free probability theory (With an appendix by Philippe Biane), Ann. of Math. (2) 149 (3) (1999), pp. 1023-1060.
  • [6] H. Bercovici and D. Voiculescu, Lévy–Hinčin type theorems for multiplicative and additive free convolution, Pacific J. Math. 153 (2) (1992), pp. 217-248.
  • [7] H. Bercovici and D. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (3) (1993), pp. 733-773.
  • [8] M. Bożejko, On Λ(p) sets with minimal constant in discrete noncommutative groups, Proc. Amer. Math. Soc. 51 (1975), pp. 407-412.
  • [9] A. Deya and I. Nourdin, Convergence of Wigner integrals to the tetilla law, ALEA 9 (2012), pp. 101-127.
  • [10] U. Franz and N. Muraki, Markov property of monotone Lévy processes, in: Infinite Dimensional Harmonic Analysis III: Proceedings of the Third German-Japanese Symposium. University of Tübingen, Germany, 15-20 September 2003, H. Heyer, T. Hirai, T. Kawazoe, and K. Saitô (Eds.), World Scientific, 2005, pp. 37-57. math.PR/0401390.
  • [11] T. Hasebe, Monotone convolution semigroups, Studia Math. 200 (2010), pp. 175-199.
  • [12] T. Hasebe and H. Saigo, The monotone cumulants, Ann. Inst. H. Poincaré Probab. Statist. 47 (4) (2011), pp. 1160-1170.
  • [13] T. Kemp, I. Nourdin, G. Peccati, and R. Speicher, Wigner chaos and the fourth moment, Ann. Probab. 40 (4) (2012), pp. 1577-1635.
  • [14] H. Maassen, Addition of freely independent random variables, J. Funct. Anal. 106 (1992), pp. 409-438.
  • [15] N. Muraki, Monotonic independence, monotonic central limit theorem and monotonic law of small numbers, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001), pp. 39-58.
  • [16] N. Muraki, The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), pp. 337-371.
  • [17] N. Muraki, Monotonic convolution and monotonic Lévy-Hincin formula, preprint, 2000.
  • [18] I. Nourdin and G. Peccati, Poisson approximations in the free Wigner chaos, Ann. Probab. 41 (4) (2013), pp. 2709-2723.
  • [19] I. Nourdin and G. Poly, Convergence in law in the second Wiener/Wigner chaos, Electron. Comm. Probab. 17 (36) (2012).
  • [20] D. Nualart and G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (1) (2005), pp. 177-193.
  • [21] R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994), pp. 611-628.
  • [22] R. Speicher and R. Woroudi, Boolean convolution, in: Free Probability Theory, D. Voiculescu (Ed.), Fields Inst. Commun. 12, Amer. Math. Soc., 1997, pp. 267-280.
  • [23] D. Voiculescu, Symmetries of some reduced free product C*-algebras, in: Operator Algebras and Their Connections with Topology and Ergodic Theory, Lecture Notes in Math., Vol. 1132, Springer, Berlin-New York 1985, pp. 556-588.
  • [24] D. Voiculescu, Addition of certain non-commutating random variables, J. Funct. Anal. 66 (1986), pp. 323-346.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-fb7767b4-ab16-4c55-b43e-787bc5e2c0c4
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