Quantitative Breuer–Major theorems

We consider sequences of random variables of the type Sn=n−1/2∑k=1n{f(Xk)−E[f(Xk)]}, n≥1n≥1, where X=(Xk)k∈ZX=(Xk)k∈Z is a dd-dimensional Gaussian process and f:Rd→Rf:Rd→R is a measurable function. It is known that, under certain conditions on ff and the covariance function rr of XX, SnSn converges in distribution to a normal variable SS. In the present paper we derive several explicit upper bounds for quantities of the type |E[h(Sn)]−E[h(S)]||E[h(Sn)]−E[h(S)]|, where hh 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 V ar[f(X1)] and on simple infinite series involving the components of rr. In particular, our results generalize and refine some classic CLTs given by Breuer and Major, Giraitis and Surgailis, and Arcones, concerning the normal approximation of partial sums associated with Gaussian-subordinated time series.