A contribution to the theory of aproximately mindmax detecting of a vector signal in a Gaussian noise

In a normal size $N$ sample of independent indentically distributed variables $X_i\in\mathbf N(\xi,\Sigma)$ with covariance matrix unknown, the hypothesis $H_0\colon\xi=0$ againts $H_\delta\colon N\xi^T\Sigma^{-1}\xi\ne\delta$ is tested.
The well known Hotelling test $(T^2)$ is proved to be approximately minimax with considerable accuracy in the following sense: for all randomized level $\alpha$ tests $\Phi$ and a fixed positive $\delta$ we have:
$$ \sup_\Phi\inf_{\theta\in H_\delta}\mathbf E_\theta\Phi-\inf_{\theta\in H_\delta}\mathbf E_\theta(T^2)=O(\exp c_1N^{1/4}\ln N) $$
with $\theta=(\xi,\Sigma),$ $c_1>0$.