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The functional Breuer–Major theorem

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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Notes

  1. Chambers and Slud criterion corresponds to Ben Hariz criterion (1.5) with \(R=\frac{3}{2}\) and without the terms \(\sum _{k\in {\mathord {\mathbb Z}}}|\rho (k)|^q\) all bounded by (1.2).

  2. Compared to [1], condition (1.5) is stated here with \(\sqrt{q!}\) instead of \((\sqrt{q!})^{-1}\) (since we work here with Hermite polynomials with leading coefficient 1) and with sums replacing integrals (since we work here in a discrete framework).

  3. The statement of [9, Prop. 5.1.5] is with \(t^{-1/2}\) instead of \(\frac{e^{-t}}{\sqrt{1-e^{-2t}}}\), but the given proof actually provides the estimate stated in (2.15).

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Acknowledgements

We thank two anonymous referees for their careful reading, their constructive remarks and their useful suggestions.

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Correspondence to Ivan Nourdin.

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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