Rate of convergence in the Breuer-Major theorem via chaos expansions

Abstract

We show new estimates for the total variation and Wasserstein distances in the framework of the Breuer-Major theorem. The results are based on the combination of Stein's method for normal approximations and Malliavin calculus together with Wiener chaos expansions. Mathematics Subject Classifications (2010): 60H15, 60H07, 60G15, 60F05.

Berry-Esseen bounds in the Breuer-Major CLT and Gebelein’s inequality

“…In view of Proposition 3, applying the upper bound in Theorem 2-(ii) to the case ϕ = H 2 (and ρ ∈ ℓ b (Z), for some 1 ≤ b < 2), one obtains a rate which is not optimal. The already mentioned reference [9] shows that our results are, in general, not optimal also for the case ϕ(x) = |x| − 2/π. Further discussions around this problem are gathered at the end of the paper -see Section 4.…”

“…Such a case is not covered by the findings of [20] or [14,15,24] (due to the lack of sufficient regularity for the function ϕ), and enters indeed the framework of our main result, stated in Theorem 2. The case of such a mapping is also covered by the recent reference [9], where convergence in total variation is deduced for a class much smaller than D 1,4 , containing however ϕ(x) = |x| − 2/π.…”

The Breuer-Major Theorem in total variation: improved rates under minimal regularity

“…We will see that our estimates imply minimal regularity conditions on g, in order for the limiting relation d TV (Y n , N ) → 0 (or, equivalently, d TV (F n , N (0, σ 2 )) → 0) to take place. Moreover, under comparable regularity assumptions on g, the rates of convergence provided by our bounds are better than or commmensurate to the best estimates to date, obtained in [9,17,23]. The main tool exploited in our analysis is a non-trivial combination of Gebelein's inequality (recalled in Section 2.4 below, and already used in [17]), and some novel estimates involving Malliavin operators -see e.g.…”

“…(2) Given g = c q H q ∈ L 2 (γ), we define A(g) := |c q |H q , that is, A(g) is the element of L 2 (γ) obtained by taking the absolute value of the coefficients appearing in the Hermite expansion of g. In [9], the following results are proved: (2a) the bound…”