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Stochastic Integral and Covariation Representations for Rectangular Lévy Process Ensembles

  • J. Armando Domínguez-Molina
  • Alfonso Rocha-Arteaga
Conference paper
Part of the Progress in Probability book series (PRPR, volume 69)

Abstract

A Bercovici-Pata bijection \(\Lambda _{c}\) from the set of symmetric infinitely divisible distributions to the set of ⊞  c -free infinitely divisible distributions, for certain free convolution ⊞  c is introduced in Benaych-Georges (Random matrices, related convolutions. Probab Theory Relat Fields 144:471–515, 2009. Revised version of F. Benaych-Georges: Random matrices, related convolutions. arXiv, 2005). This bijection is explained in terms of complex rectangular matrix ensembles whose singular distributions are ⊞  c -free infinitely divisible. We investigate the rectangular matrix Lévy processes with jumps of rank one associated to these rectangular matrix ensembles. First as general result, a sample path representation by covariation processes for rectangular matrix Lévy processes of rank one jumps is obtained. Second, rectangular matrix ensembles for ⊞  c -free infinitely divisible distributions are built consisting of matrix stochastic integrals when the corresponding symmetric infinitely divisible distributions under \(\Lambda _{c}\) admit stochastic integral representations. These models are realizations of stochastic integrals of nonrandom functions with respect to rectangular matrix Lévy processes. In particular, any ⊞  c -free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type \(\int _{0}^{\infty }e^{-t}\mathrm{d}\Psi (t)\), where \(\left \{\Psi (t): t \geq 0\right \}\) is a rectangular matrix Lévy process.

Keywords

Random matrices Rectangular random matrix model Complex matrix semimartingales Complex matrix Lévy processes Lévy measures Ornstein-Uhlenbeck rectangular type processes Infinitely divisible distribution Free infinitely divisible distribution Bercovici-Pata bijection 

Mathematics Subject Classification (2000).

46L54 60E07 60H05 15A52 60G51 60G57 

References

  1. 1.
    T. Aoyama, M. Maejima, Some classes of infinitely divisible distributions on \(\mathbb{R}^{d}\) (a survey), unpublished note (2008). Revised version of the Research Report: T. Aoyama, M. Maejima, Some classes of infinitely divisible distributions on \(\mathbb{R}^{d}\) (a survey), The Institute of Statistical Mathematics Cooperate Research Report, Tokyo, vol. 184 (2006), pp. 5–13Google Scholar
  2. 2.
    T. Aoyama, M. Maejima, Characterizations of subclasses of type G distributions on \(\mathbb{R}^{d}\) by stochastic integral representations. Bermoulli 13(1), 148–160 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Applebaum, Lévy processes and stochastic integrals in Banach spaces. Probab. Math. Stat. 27, 75–88 (2007)MathSciNetGoogle Scholar
  4. 4.
    O.E. Barndorff-Nielsen, M. Maejima, K. Sato, Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12, 1–33 (2006)MathSciNetGoogle Scholar
  5. 5.
    O.E. Barndorff-Nielsen, R. Stelzer, Positive-definite matrix processes of finite variation. Probab. Math. Stat. 27, 3–43 (2007)MathSciNetGoogle Scholar
  6. 6.
    O.E. Barndorff-Nielsen, R. Stelzer, Multivariate supOU processes. Ann. Appl. Probab. 21, 140–182 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    F. Benaych-Georges, Classical and free infinitely divisible distributions and random matrices. Ann. Probab. 33, 1134–1170 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    F. Benaych-Georges, Infinitely divisible distributions for rectangular free convolution- classification and matricial interpretation. Probab. Theory Relat. Fields 139, 143–189 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    F. Benaych-Georges, Random matrices, related convolutions. Probab. Theory Relat. Fields 144, 471–515 (2009). Revised version of F. Benaych-Georges, Random matrices, related convolutions. arXiv (2005)Google Scholar
  10. 10.
    T. Cabanal-Duvillard, A matrix representation of the Bercovici-Pata bijection. Electron. J. Probab. 10, 632–661 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.A. Domínguez-Molina, V. Pérez-Abreu, A. Rocha-Arteaga, Covariation representations for Hermitian Lévy process ensembles for free infinitely divisible distributions. Electron. Commun. Probab. 18, 1–14 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    J.A. Domínguez-Molina, A. Rocha-Arteaga, Random matrix models of stochastic integrals type for free infinitely divisible distributions. Periodica Mathematica Hungarica. 64(2), 145–160 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    F. Hiai, D. Petz, The Semicircle Law, Free Random Variables and Entropy. Mathematics Surveys and Monographs, vol. 77 (American Mathematical Society, Providence, 2000)Google Scholar
  14. 14.
    Z.J. Jurek, W. Vervaat, An integral representation for selfdecomposable Banach space valued random variables. Z. Wahrscheinlichkeitstheorie. Verw. Geb. 62, 247–262 (1983)MathSciNetCrossRefGoogle Scholar
  15. 15.
    J.F.C. Kingman, Random walks with spherical symmetry. Acta Math. 109(1), 11–53 (1963)MathSciNetCrossRefGoogle Scholar
  16. 16.
    V. Pérez-Abreu, N. Sakuma, Free generalized gamma convolutions. Electron. Commun. Probab. 13, 526–539 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    P. Protter, Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, vol. 21 (Springer, Berlin/New York, 2004)Google Scholar
  18. 18.
    B.S. Rajput, J. Rosiński, Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82, 451–487 (1989)CrossRefGoogle Scholar
  19. 19.
    A. Rocha-Arteaga, K. Sato, Topics in Infinitely Divisible Distributions and Lévy Processes. Aportaciones Matemáticas, Investigación, vol. 17 (Sociedad Matemática Mexicana, México, 2003)Google Scholar
  20. 20.
    J. Rosiński, On series representations of infinitely divisible random vectors. Ann. Probab. 18, 405–430 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    K. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, Cambridge, 1999)Google Scholar
  22. 22.
    K. Sato, Additive processes and stochastic integrals. Ill. J. Math. 50, 825–851 (2006)Google Scholar
  23. 23.
    K. Sato, M. Yamazato, Stationary processes of Ornstein-Uhlenbeck type, in Probability Theory and Mathematical Statistics, ed. by K. Ito, J.V. Prokhorov. Lecture Notes in Mathematics, vol. 1021 (Spriger, Berlin, 1983), pp. 541–551Google Scholar
  24. 24.
    K. Urbanik, W.A. Woyczynski, A random integral and Orlicz spaces. Bull. Acad. Polon. Sci. Math. Astro. e Phys. 15, 161–168 (1967)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • J. Armando Domínguez-Molina
    • 1
  • Alfonso Rocha-Arteaga
    • 1
  1. 1.Facultad de Ciencias Físico-MatemáticasUniversidad Autónoma de SinaloaSinaloaMéxico

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