Quantitative Breuer-Major Theorems
By Ivan Nourdin, Giovanni Peccati, Mark Podolskij. 2010
81 citations recorded by the source corpus.
We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n f(X_k)$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ is a measurable function. It is known that, under certain conditions on $f$ and the covariance function $r$ of $X$, $S_n$ converges in distribution to a normal variable $S$. In the present paper we derive several explicit upper bounds for quantities of the type $|\E[h(S_n)] -\E[h(S)]|$, where $h$ is a sufficiently smooth test function. Our methods are based on Malliavin calculus, on interpolation techniques and on the Stein's method for normal approximation. The bounds deduced in our paper depend only on $\E[f^2(X_1)]$ and on simple infinite series involving the components of $r$. In particular, our results generalize and refine some classic CLTs by Breuer-Major, Giraitis-Surgailis and Arcones, concerning the normal approximation of partial sums associated with Gaussian-subordinated time-series.
Subfields: Finance, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/ebb894f601202246e98638b9e86879e42457f3b0
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Citations in the corpus
- A central limit theorem for the Euler characteristic of a Gaussian excursion set (2016, methodology): “Our second tool is the Stein method, recently revisited in [25, 27, 28] for instance.”
- Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S^d (2014, unknown): “subordi-nated random fields In this section, we first briefly review some notation and the main results in this area, which we shall deeply exploit in the sequel of the paper.”
- Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants (2011, unknown): “In this final section, we determine optimal rates of convergence associated with the well-known Breuer-Major CLT for Gaussian-subordinated random sequences – see [3] for the original paper, or [14] for a more modern reference.”