Electronic Communications in Probability

Almost sure limit theorems on Wiener chaos: the non-central case

Ehsan Azmoodeh and Ivan Nourdin

Full-text: Open access

Abstract

In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Itô integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 9, 12 pp.

Dates
Received: 3 October 2018
Accepted: 18 January 2019
First available in Project Euclid: 15 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1550199821

Digital Object Identifier
doi:10.1214/19-ECP212

Mathematical Reviews number (MathSciNet)
MR3916341

Zentralblatt MATH identifier
1412.60041

Subjects
Primary: 60F05: Central limit and other weak theorems 60G22: Fractional processes, including fractional Brownian motion 60G15: Gaussian processes 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
almost sure limit theorem multiple Wiener-Itô integrals Malliavin calculus characteristic function Wiener chaos Hermite distribution fractional Brownian motion

Rights
Creative Commons Attribution 4.0 International License.

Citation

Azmoodeh, Ehsan; Nourdin, Ivan. Almost sure limit theorems on Wiener chaos: the non-central case. Electron. Commun. Probab. 24 (2019), paper no. 9, 12 pp. doi:10.1214/19-ECP212. https://projecteuclid.org/euclid.ecp/1550199821


Export citation

References

  • [1] Bercu, B., Nourdin, I., and Taqqu, M.: Almost sure central limit theorems on the Wiener space. Stoch. Proc. Appl. 120, (2010), 1607–1628.
  • [2] Berkes, I., and Csáki, E.: A universal result in almost sure central limit theory. Stoch. Process. Appl. 94, no. 1, (2001), 105–134.
  • [3] Berkes, I., and H. Dehling, H.: Some limit theorems in log density. Ann. Probab. 21, (1993), 1640–1670.
  • [4] Berkes, I., and Horváth, L.: Limit theorems for logarithmic averages of fractional Brownian motions. J. Theoret. Probab. 12, no. 4, (1999), 985–1009.
  • [5] Breton, J.-C., and Nourdin, I.: Error bounds on the non-normal approximation of Hermite variations of fractional Brownian motion. Electron. Comm. Probab. 13, (2008), 482–493.
  • [6] Breuer, P., and Major, P.: Central limit theorems for nonlinear functionals of Gaussian fields, J. Multivariate Anal. 13, no. 3, (1983), 425–441.
  • [7] Brosamler, G.A.: An almost everywhere central limit theorem. Math. Proc. Cambridge Philos. Soc. 104, no. 3, (1988), 561–574.
  • [8] Dobrushin, R.L., and Major, P.: Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 50, (1979), 27–52.
  • [9] Giraitis, l., and Surgailis, D.: CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 70, (1985), 191–212.
  • [10] Gonchigdanzan, K.: Almost Sure Central Limit Theorems. Ph.D. Thesis, University of Cincinnati, Available online, 2001.
  • [11] Ibragimov, I.A., and Lifshits, M.A.: On the convergence of generalized moments in almost sure central limit theorem. Statist. Probab. Lett. 40, no. 4, (1998), 343–351.
  • [12] Ibragimov, I.A., and Lifshits, M.A.: On limit theorems of “almost sure” type. Theory Probab. Appl. 44, no. 2, (2000), 254–272.
  • [13] Lacey, M.T., and Philipp, W.: A note on the almost sure central limit theorem. Statist. Probab. Lett. 9, (1990), 201–205.
  • [14] Lévy, P.: Théorie de l’addition des variables aléatoires. (1937), Gauthiers-Villars.
  • [15] Nourdin, I.: Malliavin-Stein approach: a webpage maintained by Ivan Nourdin. http://tinyurl.com/kvpdgcy
  • [16] Nourdin, I., Nualart, D., and Peccati, G.: Strong asymptotic independence on Wiener chaos. Proc. American Math. Society 144, no. 2, (2016), 875–886.
  • [17] Nourdin, I., and Peccati, G.: Stein’s method on Wiener chaos. Probab. Theory Related Fields 145, no. 1, (2009), 75–118.
  • [18] Nourdin, I., and Peccati, G.: Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics. Cambridge University Press, 2012.
  • [19] Nourdin, I., and Poly, G.: Convergence in total variation on Wiener chaos. Stoch. Proc. Appl. 123, (2013), 651–674.
  • [20] Nourdin, I., and Rosiński, J.: Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws. Ann. Probab. 42, no. 2, (2014), 497–526.
  • [21] Nualart, D., and Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33, no. 1, (2005), 177–193.
  • [22] Nualart, D.: The Malliavin calculus and related topics. 2nd edition. Probability and Its Applications, Springer, 2006.
  • [23] Nualart, D., and Nualart, E.: Introduction to Malliavin Calculus. Institute of Mathematical Statistics Textbooks, Cambridge University Press, 2018.
  • [24] Peligrad, M., and Shao, Q. M.: A note on the almost sure central limit theorem for weakly dependent random variables. Statist. Probab. Lett. 22, no. 2, (1995), 131–136.
  • [25] Schatte, P.: On strong versions of the central limit theorem. Math. Nachr. 137, (1988), 249–256.
  • [26] Spitzer, F.: Principles of random walk. Graduate Texts in Mathematics 34, Second edition, Springer-Verlag, New York-Heidelberg, 1976.
  • [27] M.S.Taqqu, M.S.: Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 50, (1979), 53–83.