Continuous Breuer–Major theorem: Tightness and nonstationarity

Abstract

Let Y = (Y (t)) t≥0 be a zero-mean Gaussian stationary process with covariance function ρ : R → R satisfying ρ(0) = 1. Let f : R → R be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of f is d ≥ 1. If R |ρ(s)| d ds < ∞, then the celebrated Breuer-Major theorem (in its continuous version) asserts that the finite-dimensional distributions of Z ε := √ ε ·/ε 0 f (Y (s))ds converge to those of σW as ε → 0, where W is a standard Brownian motion and σ is some explicit constant. Since its first appearance in 1983, this theorem has become a crucial probabilistic tool in different areas, for instance in signal processing or in statistical inference for fractional Gaussian processes.The goal of this paper is twofold. Firstly, we investigate the tightness in the Breuer-Major theorem. Surprisingly, this problem did not receive a lot of attention until now, and the best available condition due to Ben Hariz [1] is neither arguably very natural, nor easy-to-check in practice. In contrast, our condition very simple, as it only requires that |f | p must be integrable with respect to the standard Gaussian measure for some p strictly bigger than 2. It is obtained by means of the Malliavin calculus, in particular Meyer inequalities.Secondly, and motivated by a problem of geometrical nature, we extend the continuous Breuer-Major theorem to the notoriously difficult case of self-similar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Browninan motion concludes the paper.where H q (x) is the qth Hermite polynomial.It has become a central result in modern stochastic analysis that, under the conditionconverge, as ε tends to zero, to those of σW , where W = (W (t)) t≥0 is a standard Brownian

Oscillating Gaussian processes

“…Then, as r is integrable, continuous version of the Breuer-Major theorem (see e.g. [3]) implies the claim directly. Under the other two conditions, we first note that the only contributing factor to the limiting distribution in (3.6) is…”

“…In the case (3), the limiting process is σB H t , where B H is the fractional Brownian motion. Indeed, the last case follows from a classical result by Taqqu [12] and the first case from [3] and from the fact that all moments of X are finite. However, from practical point of view, translating these results to functional versions of the estimatorsα + (T ) andα − (T ) is not feasible.…”

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

“…Changing the exponents β jk + 1 in to β jk for (j, k) ∈ {(1, 2), (1,3), (1,4), (5,6), (5, 7), (5, 8)}, we can write…”

“…The exponents β jk induce an unordered simple graph on the set of vertices V = {1, 2, 3, 4, 5, 8} by putting an edge between j and k whenever β jk = 0. Because β 12 , β 13 ≥ 1, β 14 ≥ 1, β 56 ≥ 1, β 57 ≥ 1 and β 58 ≥ 1, there are edges connecting the pairs of vertices (1, 2), (1,3), (1,4), (5,6), (5,7) and (5,8). Condition (5.10) means that the degree of each vertex is at least 2.…”

Averaging Gaussian functionals

“…with σ 2 := ω d q≥m c 2 q q! R d ρ(x) m dx ∈ [0, ∞), ω d being the volume of B 1 ; see also [3,24]. Example 1.2.…”