Concentration inequalities and moment bounds for sample covariance operators

Let X,X₁,,Xₙ, be i.i.d. centered Gaussian random variables in a separable Banach space E with covariance operator Σ: \[Σ:E^{}↦ E,Σ u=E X,u X, u∈ E^{}.\] The sample covariance operator Σ̂:E^↦ E is defined as \[Σ̂u:=n⁻¹∑ⱼ₌₁ⁿ Xⱼ,u Xⱼ, u∈ E^{}.\] The goal of the paper is to obtain concentration inequalities and expectation bounds for the operator norm Σ̂-Σ of the deviation of the sample covariance operator from the true covariance operator. In particular, it is shown that \[E Σ̂-Σ Σ (√{{{r}(Σ)}{n}} {{r}(Σ)}{n}),\] where \[{r}(Σ):={(E X )²}{ Σ }.\] Moreover, it is proved that, under the assumption that r(Σ)≤ n, for all t≥1, with probability at least 1-e⁻ᵗ \[ Σ̂-Σ -M Σ (√t/n t/n),\] where M is either the median, or the expectation of Σ̂-Σ. On the other hand, under the assumption that r(Σ)≥ n, for all t≥1, with probability at least 1-e⁻ᵗ \[ Σ̂-Σ -M Σ (√{{{r}(Σ)}{n}}√t/n t/n).\]