Thinning a Wishart random matrix

Summary Recent work has explored data thinning, a generalization of sample splitting that involves decomposing a (possibly matrix-valued) random variable into independent components. In the special case of an $ n\times p $ random matrix with independent and identically distributed $ N_{p}(\mu,\Sigma) $ rows, Dharamshi et al. (2026)provided a comprehensive analysis of the settings in which thinning is or is not possible: briefly, if $ \Sigma $ is unknown then one can thin provided that $ n \gt 1 $. However, in some situations a data analyst may have access only to summary statistics of the data, e.g., due to privacy considerations. While the sample mean follows a Gaussian distribution, the sample covariance follows, up to scaling, a Wishart distribution, for which no thinning strategies have yet been proposed. In this note, we fill this gap: we show that it is possible to generate two or more independent data matrices with independent $ N_{p}(\mu,\Sigma) $ rows, based only on the sample mean and sample covariance matrix. These independent data matrices can either be used directly within a train-test paradigm or be used to derive independent summary statistics. Furthermore, they can be recombined to yield the original sample mean and sample covariance. The key insight that enables this development is an algorithm that decomposes a Wishart random matrix into a matrix square root with independent and identically distributed Gaussian rows.