Sharp minimax tests for large Toeplitz covariance matrices with repeated observations

We observe a sample of $n$ independent $p$-dimensional Gaussian vectors with Toeplitz covariance matrix $\Sigma = [\sigma_{|i-j|}]_{1 \leq i,j \leq p}$ and $\sigma_0=1$. We consider the problem of testing the hypothesis that $\Sigma$ is the identity matrix asymptotically when $n \to \infty$ and $p \to \infty$. We suppose that the covariances $\sigma_k$ decrease either polynomially ($\sum_{k \geq 1} k^{2\alpha} \sigma^2_{k} \leq L$ for $\alpha >1/4$ and $L>0$) or exponentially ($\sum_{k \geq 1} e^{2Ak} \sigma^2_{k} \leq L$ for $A,L>0$). We consider a test procedure based on a weighted U-statistic of order 2, with optimal weights chosen as solution of an extremal problem. We give the asymptotic normality of the test statistic under the null hypothesis for fixed $n$ and $p \to + \infty$ and the asymptotic behavior of the type I error probability of our test procedure. We also show that the maximal type II error probability, either tend to $0$, or is bounded from above. In the latter case, the upper bound is given using the asymptotic normality of our test statistic under alternatives close to the separation boundary. Our assumptions imply mild conditions: $n=o(p^{2\alpha - 1/2})$ (in the polynomial case), $n=o(e^p)$ (in the exponential case). We prove both rate optimality and sharp optimality of our results, for $\alpha >1$ in the polynomial case and for any $A>0$ in the exponential case. A simulation study illustrates the good behavior of our procedure, in particular for small $n$, large $p$.