Let $X_1,X_2, \ldots$ be independent and identically distributed random elements taking values in a separable Hilbert space $\mathbb{H}$. With applications for functional data in mind, $\mathbb{H}$ may be regarded as a space of square-integrable functions, defined on a compact interval. We propose and study a novel test of the hypothesis $H_0$ that $X_1$ has some unspecified non-degenerate Gaussian distribution. The test statistic $T_n=T_n(X_1,\ldots,X_n)$ is based on a measure of deviation between the empirical characteristic functional of $X_1,\ldots,X_n$ and the characteristic functional of a suitable Gaussian random element of $\mathbb{H}$. We derive the asymptotic distribution of $T_n$ as $n \to \infty$ under $H_0$ and provide a consistent bootstrap approximation thereof. Moreover, we obtain an almost sure limit of $T_n$ as well as a normal limit distribution of $T_n$ under alternatives to Gaussianity. Simulations show that the new test is competitive with respect to the hitherto few competitors available.

中文翻译：

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通过经验特征泛函检验希尔伯特空间中的高斯性

令 $X_1,X_2,\ldots$ 是独立同分布的随机元素，取值在可分离的希尔伯特空间 $\mathbb{H}$ 中。考虑到函数数据的应用，$\mathbb{H}$ 可以被视为一个平方可积函数空间，定义在一个紧凑的区间上。我们提出并研究了假设 $H_0$ 的新测试，即 $X_1$ 具有一些未指定的非退化高斯分布。检验统计量 $T_n=T_n(X_1,\ldots,X_n)$ 基于 $X_1,\ldots,X_n$ 的经验特征泛函与 $\ 的合适高斯随机元素的特征泛函之间的偏差度量mathbb{H}$. 我们将 $T_n$ 的渐近分布推导出为 $n \to \infty$ 在 $H_0$ 下，并提供其一致的引导程序近似值。而且，我们获得了几乎确定的 $T_n$ 极限以及 $T_n$ 在高斯替代方案下的正态极限分布。模拟表明，相对于迄今为止可用的少数竞争对手，新测试具有竞争力。