Let $E$ be a separable Banach space and let $X, X_1,\dots, X_n, \dots$ be i.i.d. Gaussian random variables taking values in $E$ with mean zero and unknown covariance operator $\Sigma: E^{\ast}\mapsto E.$ The complexity of estimation of $\Sigma$ based on observations $X_1,\dots, X_n$ is naturally characterized by the so called effective rank of $\Sigma:$ ${\bf r}(\Sigma):= \frac{{\mathbb E}_{\Sigma}\|X\|^2}{\|\Sigma\|},$ where $\|\Sigma\|$ is the operator norm of $\Sigma.$ Given a smooth real valued functional $f$ defined on the space $L(E^{\ast},E)$ of symmetric linear operators from $E^{\ast}$ into $E$ (equipped with the operator norm), our goal is to study the problem of estimation of $f(\Sigma)$ based on $X_1,\dots, X_n.$ The estimators of $f(\Sigma)$ based on jackknife type bias reduction are considered and the dependence of their Orlicz norm error rates on effective rank ${\bf r}(\Sigma),$ the sample size $n$ and the degree of H\"older smoothness $s$ of functional $f$ are studied. In particular, it is shown that, if ${\bf r}(\Sigma)\lesssim n^{\alpha}$ for some $\alpha\in (0,1)$ and $s\geq \frac{1}{1-\alpha},$ then the classical $\sqrt{n}$-rate is attainable and, if $s> \frac{1}{1-\alpha},$ then asymptotic normality and asymptotic efficiency of the resulting estimators hold. Previously, the results of this type (for different estimators) were obtained only in the case of finite dimensional Euclidean space $E={\mathbb R}^d$ and for covariance operators $\Sigma$ whose spectrum is bounded away from zero (in which case, ${\bf r}(\Sigma)\asymp d$).

翻译：设$E$为可分Banach空间，$X, X_1,\dots, X_n, \dots$是取值于$E$的独立同分布高斯随机变量，均值为零且具有未知协方差算子$\Sigma: E^{\ast}\mapsto E.$ 基于观测$X_1,\dots, X_n$估计$\Sigma$的复杂度自然由其所谓有效秩刻画：${\bf r}(\Sigma):= \frac{{\mathbb E}_{\Sigma}\|X\|^2}{\|\Sigma\|},$ 其中$\|\Sigma\|$为$\Sigma$的算子范数。给定定义在从$E^{\ast}$到$E$的对称线性算子空间$L(E^{\ast},E)$（配备算子范数）上的光滑实值泛函$f$，本文研究基于$X_1,\dots, X_n$估计$f(\Sigma)$的问题。考虑基于折刀型偏差缩减的$f(\Sigma)$估计量，并研究其Orlicz范数误差率对有效秩${\bf r}(\Sigma)$、样本量$n$以及泛函$f$的Hölder光滑度$s$的依赖性。特别地，结果表明：若存在$\alpha\in (0,1)$使得${\bf r}(\Sigma)\lesssim n^{\alpha}$且$s\geq \frac{1}{1-\alpha}$，则可达到经典$\sqrt{n}$速率；若$s> \frac{1}{1-\alpha}$，则所得估计量具有渐近正态性和渐近有效性。此前，此类结果（针对不同估计量）仅对有限维欧氏空间$E={\mathbb R}^d$且协方差算子$\Sigma$谱远离零（此时${\bf r}(\Sigma)\asymp d$）的情形成立。