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Preliminaries of Lévy Processes

  • Bronius Grigelionis
Chapter
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Abstract

Let \((\Omega , \fancyscript{F}, \mathrm P )\) be a probability space and \((R^d, \fancyscript{B}(R^d), \langle \cdot ,\cdot \rangle )\) be a d-dimensional Euclidean space \(R^d\) with the \(\sigma \)-algebra of Borel subsets \(\fancyscript{B}(R^d)\), the scalar product \(\langle x,y\rangle =\sum\nolimits ^{d}_{j=1}x_jy_j\) for row vectors \(x=(x_1,\ldots ,x_d)\), \(y=(y_1,\ldots ,y_d)\) and the norm \(|x|=\sqrt{\langle x,x\rangle }\).

Keywords

Probability Space Sample Path Borel Subset Polar Decomposition Divisible Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Appelbaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)Google Scholar
  2. 2.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)Google Scholar
  3. 3.
    Skorohod, A.V.: Random Processes with Independent Increments. Kluwer, Amsterdam (1991)Google Scholar
  4. 4.
    Rocha-Arteaga, A., Sato, K.: Topics in Infinitely Divisible Distributions and Lévy Processes. Sociedad Matematática Mexicana, Mexico (2003)Google Scholar
  5. 5.
    Barndorff-Niesen, O.E., Pedersen, J., Sato, K.: Multivariate subordination, self-decomposability and stability. Adv. Prob. 33, 160–187 (2001)Google Scholar
  6. 6.
    Grigelionis, B.: Thorin classes of Lévy processes and their transforms. Lith. Math. J. 48(3), 294–315 (2008)Google Scholar
  7. 7.
    Grigelionis, B.: On subordinated multivariate Gaussian Lévy processes. Acta Appl. Math. 96, 233–249 (2007)Google Scholar
  8. 8.
    Sato, K., Yamazato, M.: Stationary processes of Ornstein-Uhlenbeck type. In: Probability Theory and Mathematical Statististics (Tbilisi, 1982), Lecture Notes in Mathematics, vol. 1021, pp. 541–551. Springer. Berlin (1983)Google Scholar
  9. 9.
    Bertoin, J.: Lévy Processes, Caml. Tracts. Math. 121, Cambridge University Press, Cambridge, (1996)Google Scholar
  10. 10.
    Bertoin, J.: Subordinators: Examples and applications. In: Ecole d’Ete de Probabilités de Saint - Flour XXVII, Lecture Notes in Math., vol. 1727, pp. 1–91. Springer. Heidelberg, (1999)Google Scholar
  11. 11.
    Bondesson, L.: Generalized gamma convolutions and related classes of distributions and densities. In: Lecture Notes in Statistics, vol. 76. Springer-Verlag, Berlin (1992)Google Scholar
  12. 12.
    Schilling, R., Song, R., Vondraček, Z.: Bernstein functions-theory and applications. De Gruyter (2010)Google Scholar
  13. 13.
    Jørgensen, B.: The Theory of Dispersion Models, Monographs on Statistics and Applied Probability, Vol. 76, Chapman & Hall, London (1997)Google Scholar
  14. 14.
    Tweedie, M.C.K.: An index which distinguishes between some important exponential families. In: Statistics: Applications and New Directions (Calcutta, 1981), Indian Statist. Inst., Calcutta, pp. 579–604 (1984)Google Scholar
  15. 15.
    Vinogradov, V.: Properties of certain Lévy and geometric Lévy processes. Commun. Stoch. Anal. 2(2), 193–208 (2008)Google Scholar
  16. 16.
    Grigelionis, B.: Extending the Thorin class. Lith. Math. J. 51(2), 194–206 (2011)Google Scholar
  17. 17.
    Goldie, C.: A class of infinitely divisible random variables. Proc. Cambridge Philos. Soc. 63, 1141–1143 (1967)Google Scholar
  18. 18.
    Steutel, F.W.: Note on the infinite divisibility of exponential mixtures. Ann. Math. Statist. 38, 1303–1305 (1967)Google Scholar
  19. 19.
    Steutel, F.W., Van Harn, K.: Infinite divisibility of probability distributions on the real line. Monographs and Textbooks in Pure and Appl. Math., vol. 259. Marcel Dekker, New York (2004)Google Scholar
  20. 20.
    Fisher, R.A.: The general sampling distribution of the multiple correlation coefficient. Proc. Royal Soc. London 121A, 654–673 (1928)Google Scholar
  21. 21.
    Alam, K., Saxena, L.: Estimation of the noncentrality parameter of a chi-square distribution. Ann. Statist. 10, 1012–1016 (1982)Google Scholar
  22. 22.
    Johnson, N.L., Kotz, S.: Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York (1972)Google Scholar
  23. 23.
    Bochner, S.: Diffusion Equation and Stochastic Processes. Proc. Nat. Acad. Sci. USA 35, 368–370 (1949)Google Scholar
  24. 24.
    Bochner, S.: Harmonic analysis and theory of probability. Univ. California Press, Berkeley and Los Angeles (1955)Google Scholar
  25. 25.
    Zolotarev, V.M.: Distribution of the Superposition of Infinitely Divisible Processes. Probab. Theory Appl. 3, pp. 185–188 (1958)Google Scholar
  26. 26.
    Ikeda, N., Watanabe, S.: On some relations between the harmonic measure and the L{\'e}vy measure for certain class of Markov processes. J. Math. Kyoto Univ. \textbf{2}, 79–95 (1962)Google Scholar
  27. 27.
    Rogozin. B.A.: On some class of processes with independent increments. Theory Probal. Appl. 10, pp. 479–483 (1965)Google Scholar
  28. 28.
    Feller, W.: An Introduction to Ptheory and its Applications. Vol. 2, 2nd edn. Wiley, New York (1971)Google Scholar

Copyright information

© The Author(s) 2013

Authors and Affiliations

  1. 1.University of VilniusVilniusLithuania

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