New asymptotic results in principal component analysis

Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space $\mathbb{H}$ with covariance operator $\mathrm{\Sigma }:=\mathbb{E}(X\otimes X)\mathrm{.}$ Let $\mathrm{\Sigma }={\sum }_{r\ge 1}{\mu }_r{P}_r$ be the spectral decomposition of $\mathrm{\Sigma }$ with distinct eigenvalues ${\mu }_1>{\mu }_2>\dots$ and the corresponding spectral projectors $P_1,P_2,\dots \mathrm{.}$ Given a sample $X_1,\dots ,X_n$ of size $n$ of i.i.d. copies of $X,$ the sample covariance operator is defined as $\hat \Sigma_n := n^{-1}\sum_{j=1}^n X_j\otimes X_j.$ The main goal of principal component analysis is to estimate spectral projectors $P_1,P_2,\dots$ by their empirical counterparts ${\widehat{P}}_1,{\widehat{P}}_2,\dots$ properly defined in terms of spectral decomposition of the sample covariance operator $\hat \Sigma_n.$ The aim of this paper is to study asymptotic distributions of important statistics related to this problem, in particular, of statistic $\mathrm{\parallel }{\widehat{P}}_r-P_r{\mathrm{\parallel }}_2^2,$ where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2^2$ is the squared Hilbert–Schmidt norm. This is done in a “high-complexity” asymptotic framework in which the so called effective rank ${\bf r}(\Sigma):=\frac{{\rm tr}(\Sigma)}{|\Sigma|{\infty}}$(${\rm tr}(\cdot)$beingthetraceand$|\cdot|{\infty}$beingtheoperatornorm)ofthetruecovariance$\Sigma$isbecominglargesimultaneouslywiththesamplesize$n,$but${\bf r}(\Sigma)=o(n)$as$n\to\infty.$ In this setting, we prove that, in the case of one-dimensional spectral projector $P_r,$ the properly centered and normalized statistic $\mathrm{\parallel }{\widehat{P}}_r-P_r{\mathrm{\parallel }}_2^2$ with {\it data-dependent} centering and normalization converges in distribution to a Cauchy type limit. The proofs of this and other related results rely on perturbation analysis and Gaussian concentration.