Continuous Breuer–Major theorem: Tightness and nonstationarity
Continuous Breuer–Major theorem: Tightness and nonstationarity
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DOI：
10.1214/19-aop1357
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发表时间：
2018-07
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影响因子：
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通讯作者：
Simon Campese;I. Nourdin;D. Nualart
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作者：
Simon Campese;I. Nourdin;D. Nualart
令 $Y=(Y(t))_{t\geq0}$ 为零均值高斯平稳过程，其协方差函数 $\rho:\mathbb{R}\to\mathbb{R}$ 满足 $\rho(0)=1$。令 $f:\mathbb{R}\to\mathbb{R}$ 为关于标准高斯测度的平方可积函数，并假设 $f$ 的 Hermite 秩为 $d\geq 1$。如果 $\int_\mathbb{R} |\rho(s)|^dds<\infty$，则著名的布洛尔大定理（连续版本）断言 $Z_\varepsilon:=\sqrt{\varepsilon}\int_0^{\cdot/\varepsilon}f(Y(s))ds$ 的有限维分布收敛于 $\sigma W$ 的有限维分布： $\varepsilon\to 0$，其中$W$是标准布朗运动，$\sigma$是一些显式常数。自 1983 年首次出现以来，该定理已成为不同领域的重要概率工具，例如信号处理或分数高斯过程的统计推断。本文的目标是双重的。首先，我们研究 Breuer-Major 定理的紧性。令人惊讶的是，这个问题直到现在才受到很多关注，而且 Ben Hariz [1] 提出的最佳可用条件既不是很自然，也不是在实践中易于检查。相比之下，我们的条件非常简单，因为它只要求 $|f|^p$ 对于严格大于 2 的 $p$ 必须相对于标准高斯测度可积。它是通过 Malliavin 微积分，特别是 Meyer 不等式获得的。其次，受几何性质问题的启发，我们将连续布洛尔-梅杰定理扩展到众所周知的困难情况，即不一定是平稳的自相似高斯过程。本文总结了与双分数布朗运动正则化版本的长度过程相关的波动的应用。
Let $Y=(Y(t))_{t\geq0}$ be a zero-mean Gaussian stationary process with covariance function $\rho:\mathbb{R}\to\mathbb{R}$ satisfying $\rho(0)=1$. Let $f:\mathbb{R}\to\mathbb{R}$ be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of $f$ is $d\geq 1$. If $\int_\mathbb{R} |\rho(s)|^dds<\infty$, then the celebrated Breuer-Major theorem (in its continuous version) asserts that the finite-dimensional distributions of $Z_\varepsilon:=\sqrt{\varepsilon}\int_0^{\cdot/\varepsilon}f(Y(s))ds$ converge to those of $\sigma W$ as $\varepsilon\to 0$, where $W$ is a standard Brownian motion and $\sigma$ 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.