Electronic Journal of Probability

Stable convergence of generalized $L^2$ stochastic integrals and the principle of conditioning

Peccati Giovanni and Murad Taqqu

Full-text: Open access

Abstract

We consider generalized adapted stochastic integrals with respect to independently scattered random measures with second moments, and use a decoupling technique, formulated as a «principle of conditioning», to study their stable convergence towards mixtures of infinitely divisible distributions. The goal of this paper is to develop the theory. Our results apply, in particular, to Skorohod integrals on abstract Wiener spaces, and to multiple integrals with respect to independently scattered and finite variance random measures. The first application is discussed in some detail in the final sectionof the present work, and further extended in a companion paper (Peccati and Taqqu (2006b)). Applications to the stable convergence (in particular, central limit theorems) of multiple Wiener-Itô integrals with respect to independently scattered (and not necessarily Gaussian) random measures are developed in Peccati and Taqqu (2006a, 2007). The present work concludes with an example involving quadratic Brownian functionals.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 15, 447-480.

Dates
Accepted: 13 April 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818486

Digital Object Identifier
doi:10.1214/EJP.v12-404

Mathematical Reviews number (MathSciNet)
MR2299924

Zentralblatt MATH identifier
1139.60024

Subjects
Primary: 60G60: Random fields
Secondary: 60G57: Random measures 60F05: Central limit and other weak theorems

Keywords
Generalized stochastic integrals Independently scattered measures Decoupling Principle of conditioning Resolutions of the identity Stable convergence Weak convergence multiple Poisson integrals Skorohod integrals

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Giovanni, Peccati; Taqqu, Murad. Stable convergence of generalized $L^2$ stochastic integrals and the principle of conditioning. Electron. J. Probab. 12 (2007), paper no. 15, 447--480. doi:10.1214/EJP.v12-404. https://projecteuclid.org/euclid.ejp/1464818486


Export citation

References

  • Billingsley P. (1969). Convergence of probability measures. Birkhäuser.
  • Brodskii M.S. (1971), Triangular and Jordan Representations of Linear Operators. Transl. Math. Monographs 32, AMS, Providence.
  • Cohen S. and Taqqu M. (2004). Small and large scale behavior of the Poissonized Telecom process. Methodology and Computing in Applied Probability 6, 363-379.
  • Engel D.D. (1982). The multiple stochastic integral. Mem. Am. Math. Society 38, 1-82.
  • Feigin P. D. (1985). Stable convergence of semimartingales. Stochastic Processes and their Applications 19, 125-134.
  • Giné E. and de la Pena V.H. (1999). Decoupling. Springer Verlag.
  • Jacod J. (1984). Une généralisation des semimartingales : les processus admettant un processus à accroissements indépendants tangent. In: Séminaire de probabilités XVIII, 91-118. LNM 1059, Springer Verlag.
  • Jacod J. (2003). On processes with conditional independent increments and stable convergence in law. In: Séminaire de probabilités XXXVI, 383-401. LNM 1801, Springer Verlag.
  • Jacod J., Klopotowski A. and Mémin J. (1982). Théorème de la limite centrale et convergence fonctionnelle vers un processus à accroissements indépendants : la méthode des martingales. Annales de l'I.H.P. Section B, 1, 1-45.
  • Jacod J. and Sadi H. (1997). Processus admettant un processus à accroissements indèpendants tangent : cas général. In: Séminaire de Probabilités XXI, 479-514. LNM 1247, Springer Verlag.
  • Jacod J. and Shiryaev A.N. (1987). Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin.
  • Jakubowski A. (1986). Principle of conditioning in limit theorems for sums of random variables. The Annals of Probability 11(3), 902-915.
  • Kabanov Y. (1975). On extended stochastic integrals. Theory of Probability and its Applications 20, 710-722.
  • Kwapien S. and Woyczynski W.A. (1991). Semimartingale integrals via decoupling inequalities and tangent processes. Probability and Mathematical Statisitics 12(2), 165-200.
  • Kwapien S. and Woyczynski W.A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser.
  • Lipster R.Sh. and Shiryaev A.N. (1980). A functional central limit theorem for semimartingales. Theory of Probability and Applications XXV, 667-688.
  • Nualart D. (1995). The Malliavin Calculus and related topics. Springer Verlag.
  • Nualart D. (1998). Analysis on Wiener space and anticipating stochastic calculus. In: Lectures on Probability Theory and Statistics. École de probabilités de St. Flour XXV (1995), 123-227. LNM 1690, Springer Verlag.
  • Nualart D. and Peccati G. (2005). Central limit theorems for sequences of multiple stochastic integrals. The Annals of Probability 33(1), 177-193.
  • Nualart D. and Schoutens W. (2000). Chaotic and predictable representation for Lévy processes. Stochastic Processes and their Applications 90, 109-122.
  • Nualart D. and J. Vives J. (1990). Anticipative calculus for the Poisson space based on the Fock space. In: Séminaire de Probabilités XXIV, 154-165 LNM 1426, Springer Verlag.
  • Peccati G. and Prünster I. (2006). Linear and quadratic functionals of random hazard rates: an asymptotic analysis. Preprint.
  • Peccati G. and Taqqu M.S. (2006a). Central limit theorems for double Poisson integrals. Preprint.
  • Peccati G. and Taqqu M.S. (2006b). Stable convergence of multiple Wiener-Itô integrals. Preprint.
  • Peccati G. and Taqqu M.S. (2007). Limit theorems for multiple stochastic integrals. Preprint.
  • Peccati G., Thieullen M. and Tudor C.A. (2006). Martingale structure of Skorohod integral processes. The Annals of Probability 34(3), 1217-1239.
  • Peccati G. and Tudor C.A. (2004). Gaussian limits for vector-valued multiple stochastic integrals. In: Séminaire de Probabilités XXXVIII, 247-262. LNM 1857, Springer Verlag.
  • Peccati G. and Yor M. (2004). Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge. In: Asymptotic Methods in Stochastics, 75-87. Fields Institute Communication Series, AMS.
  • Protter P. (1992). Stochastic Integration and Differential Equation. Springer Verlag, Berlin.
  • Rajput B.S. and Rosinski J. (1989). Spectral representation of infinitely divisble processes. Probability Theory and Related Fields 82, 451-487.
  • Revuz D. and Yor M. (1999). Continuous martingales and Brownian motion. Springer Verlag, Berlin.
  • Rota G.-C. and Wallstrom C. (1997). Stochastic integrals: a combinatorial approach. The Annals of Probability 25(3), 1257-1283.
  • Samorodnitsky G. and Taqqu M.S. (1994). Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance. Chapman and Hall. New York, London.
  • Sato K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge University Press.
  • Schreiber M. (1969). Fermeture en probabilité de certains sous-espaces d'un espace L². Zeitschrift W. v. G. 14, 36-48.
  • Surgailis D. (2003). CLTs for Polynomials of Linear Sequences: Diagram Formulae with Applications. In: Long Range Dependence, 111-128. Birkhäuser.
  • Surgailis D. (2003). Non-CLT's: U-Statistics, Multinomial Formula and Approximations of Multiple Wiener-Itô integrals. In: Long Range Dependence, 129-142. Birkhäuser.
  • Üstünel A.S. and Zakai M. (1997). The Construction of Filtrations on Abstract Wiener Space. Journal of Functional Analysis 143, 10-32.
  • Wu L.M. (1990). Un traitement unifié de la représentation des fonctionnelles de Wiener. In: Séminaire de Probabilités XXIV, 166-187. LNM 1426, Springer Verlag.
  • Xue X.-H. (1991). On the principle of conditioning and convergence to mixtures of distributions for sums of dependent random variables. Stochastic Processes and their Applications 37(2), 175-186.
  • Yosida K. (1980). Functional analysis. Springer Verlag, Berlin.