= Empirical covariance operator
{title2=$\widehat C$}
For <Hilbert-space-valued random variables> $X_1,\ldots,X_n$ and a fixed deterministic centre $m$, the <empirical covariance operator> is $\widehat C_m=n^{-1}\sum_i(X_i-m)\otimes(X_i-m)$. If the observations share mean $\mu$ and <covariance> $C$, then $\mathbb E\widehat C_m=C+(\mu-m)\otimes(\mu-m)$. This follows by expanding each centred <vector> and cancelling the mean-zero cross terms. A fitted random centre changes that expectation: for an <IID sample> centred at its <sample mean>, $\mathbb E\widehat C_{\overline X}=(n-1)C/n$. Multiplying by $n/(n-1)$ gives the usual unbiased estimate when $n>1$.
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