Abstract
Let \(X=\{ X_n\}_{n\in \mathbb {Z}}\) be zero-mean stationary Gaussian sequence of random variables with covariance function \(\rho \) satisfying \(\rho (0)=1\). Let \(\varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) be a function such that \(\mathbb {E}[\varphi (X_0)^2]<\infty \) and assume that \(\varphi \) has Hermite rank \(d \ge 1\). The celebrated Breuer–Major theorem asserts that, if \(\sum _{r\in {\mathbb {Z}}} |\rho (r)|^d<\infty \) then the finite dimensional distributions of \(\frac{1}{\sqrt{n}}\sum _{i=0}^{\lfloor n\cdot \rfloor -1} \varphi (X_i)\) converge to those of \(\sigma \,W\), where W is a standard Brownian motion and \(\sigma \) is some (explicit) constant. Surprisingly, and despite the fact this theorem has become over the years a prominent tool in a bunch of different areas, a necessary and sufficient condition implying the weak convergence in the space \(\mathbf{D}([0,1])\) of càdlàg functions endowed with the Skorohod topology is still missing. Our main goal in this paper is to fill this gap. More precisely, by using suitable boundedness properties satisfied by the generator of the Ornstein–Uhlenbeck semigroup, we show that tightness holds under the sufficient (and almost necessary) natural condition that \(\mathbb {E}[|\varphi (X_0)|^{p}]<\infty \) for some \(p>2\).
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References
Ben Hariz, S.: Limit theorems for the non-linear functionals of stationary Gaussian processes. J. Multivar. Anal. 80, 191–216 (2002)
Billingsley, P.: Convergence of Probability Measures. Wiley, Hoboken (1968)
Breuer, P., Major, P.: Central limit theorems for non-linear functionals of Gaussian fields. J. Multivar. Anal. 13, 425–441 (1983)
Chambers, D., Slud, E.: Central limit theorems for nonlinear functionals of stationary Gaussian processes. Probab. Theory Relat. Fields 80, 323–346 (1989)
Jaramillo, A., Nualart, D.: Functional limit theorem for the self-intersection local time of the fractional Brownian motion. Ann. Inst. H. Poincaré (2018) (to appear)
Meyer, P.A.: Transformations de Riesz pour les lois gaussiennes. In: Seminar on Probability, XVIII. Lecture Notes in Mathematics, vol. 1059, pp. 179–193. Springer, Berlin (1984)
Nourdin, I., Peccati, G.: Normal Approximations with Malliavin calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics, vol. 192. Cambridge University Press, Cambridge (2012)
Nualart, D.: The Malliavin Calculus and Related Topics. Probability and Its Applications, 2nd edn. Springer, Berlin (2006)
Nualart, D., Nualart, E.: Introduction to Malliavin Calculus. Institute of Mathematical Statistics Textbooks. Cambridge University Press, Cambridge (2018)
Taqqu, M.: Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. verw. Geb. 50, 53–83 (1979)
Acknowledgements
We thank two anonymous referees for their careful reading, their constructive remarks and their useful suggestions.
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David Nualart was supported by the NSF Grant DMS 1811181.
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Nourdin, I., Nualart, D. The functional Breuer–Major theorem. Probab. Theory Relat. Fields 176, 203–218 (2020). https://doi.org/10.1007/s00440-019-00917-1
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DOI: https://doi.org/10.1007/s00440-019-00917-1
Mathematics Subject Classification
- 60F17
- 60H07