Limit theorems for singular Skorohod integrals
Authors:
Denis Bell, Raul Bolaños and David Nualart
Journal:
Theor. Probability and Math. Statist. 102 (2020), 21-44
MSC (2020):
Primary 60H05, 60H07
DOI:
https://doi.org/10.1090/tpms/1126
Published electronically:
March 29, 2021
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Additional Information
Abstract: In this paper we prove the convergence in distribution of sequences of Itô and Skorohod integrals with integrands having singular asymptotic behavior. These sequences include stochastic convolutions and generalize the example $\sqrt n\int _0^1 t^n B_tdB_t$ first studied by Peccati and Yor in 2004.
References
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- Ivan Nourdin and David Nualart, Central limit theorems for multiple Skorokhod integrals, J. Theoret. Probab. 23 (2010), no. 1, 39–64. MR 2591903, DOI https://doi.org/10.1007/s10959-009-0258-y
- Ivan Nourdin, David Nualart, and Giovanni Peccati, Quantitative stable limit theorems on the Wiener space, Ann. Probab. 44 (2016), no. 1, 1–41. MR 3456331, DOI https://doi.org/10.1214/14-AOP965
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- Jichuan Wang and Stephanie T. Lanza, Preface [Special issue: Methods and applications of mixture models], Adv. Appl. Stat. Sci. 3 (2010), no. 2, 1–6. MR 2867292
- Ivan Nourdin and Guillaume Poly, Convergence in total variation on Wiener chaos, Stochastic Process. Appl. 123 (2013), no. 2, 651–674. MR 3003367, DOI https://doi.org/10.1016/j.spa.2012.10.004
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References
- D. Bell. The Malliavin calculus, Dover Publications, 2006. MR 2250060
- J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, Springer, Berlin, 1987. MR 959133
- I. Nourdin and D. Nualart. Central limit theorems for multiple Skorohod integrals. J. Theoret. Probab. 23 (2010), no. 1, 39–64. MR 2591903
- I. Nourdin, D. Nualart and G Peccati. Quantitative stable limit theorems on the Wiener space. Ann. Probab. 44 (2016), no. 1, 1–41. MR 3456331
- D. Nualart. The Malliavin calculus and related topics. Second edition, Springer, Berlin, 2006. MR 2200233
- L. Pratelli and P. Rigo. Total variation bounds for Gaussian functionals. Stoch. Proc. Appl. 129 (2019), 2231–2248. MR 3958431
- L. Pratelli and P. Rigo. Convergence in total variation to a mixture of Gaussian laws. Mathematics, Special Issue Stochastic Processes with Applications, 6, 99, 2018. MR 2867292
- I. Nourdin and G. Poly. Convergence in total variation on Wiener Chaos. Stochastic Process. Appl. 123 (2013), 651–674. MR 3003367
- G. Peccati and M.S. Taqqu. Stable convergence of multiple Wiener-Itô integrals. J. Theoret. Probab. 21 (2008), no. 3, 527–570. MR 2425357
- G. Peccati and M. Yor. Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge. In: Asymptotic Methods in Stochastics, AMS, Fields Institute Communication Series, 75–87, 2004. MR 2106849
- R. L. Wheeden and A. Zygmund. Measure and integral an Introduction to real analysis. Second Edition. MR 3381284
- P. Billingsley. Convergence of Probability Measures. Second Edition. MR 1700749
- G.D Prato and J Zabczyk. Stochastic equations in infinite dimensions. Second edition, Cambridge University Press, 2014. MR 3236753
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Additional Information
Denis Bell
Affiliation:
University of North Florida, Department of Mathematics, Jacksonville, Florida
MR Author ID:
33985
Email:
dbell@unf.edu
Raul Bolaños
Affiliation:
University of Kansas, Department of Mathematics, Lawrence, Kansas
Email:
rbolanos@ku.edu
David Nualart
Affiliation:
University of Kansas, Department of Mathematics, Lawrence, Kansas
MR Author ID:
132560
Email:
nualart@ku.edu
Keywords:
Skorohod integral,
Malliavin calculus,
convergence in law,
stochastic convolution
Received by editor(s):
September 30, 2019
Published electronically:
March 29, 2021
Additional Notes:
The third author was supported by NSF Grant DMS 1811181.
Article copyright:
© Copyright 2020
Taras Shevchenko National University of Kyiv
