Let $\Sigma$ be the covariance matrix of a multivariate normal distribution. We also know that, $$T \sim N(0,2/n) \hspace{1cm} i = j $$ $$T \sim N(0,1/n) \hspace{1cm} i \neq j $$
Is the product
$$\Sigma^{1/2}T \Sigma{1/2}$$
also multivariate normally distributed ? I guess yes but cant prove it.