The covariance matrix can be written as $$\Sigma = AA^{T}$$.
My lecturer told me we can decompose the empirical covariance matrix as
$$\widehat{\Sigma} = \Sigma + \Sigma^{1/2} T \Sigma^{1/2}$$
where $$T_{ij} \sim N(0,2/n) \hspace{1cm} i=j$$
and $$T_{ij} \sim N(0,1/n) \hspace{1cm} i \neq j$$
Do you know why this decomposition is possible ?