We consider the detection problem of correlations in a p-dimensional Gaussian vector, when we observe n independent, identically distributed random vectors, for n and p large.We assume that the covariance matrix varies in some ellipsoid with parameter α > 1/2 and total energy bounded by L > 0.We propose a test procedure based on a U-statistic of order 2 which is weighted in an optimal way. The weights are the solution of an optimization problem, they are constant on each diagonal and non-null only for the T first diagonals, where T = o(p). We show that this test statistic is asymptotically Gaussian distributed under the null hypothesis and also under the alternative hypothesis for matrices close to the detection boundary.We prove upper bounds for the total error probability of our test procedure, for α > 1/2 and under the assumption T = o(p) which implies that n = o(p 2α ). We illustrate via a numerical study the behavior of our test procedure.Moreover, we prove lower bounds for the maximal type II error and the total error probabilities. Thus we obtain the asymptotic and the sharp asymptotically minimax separation rate ϕ = (C(α, L)n 2 p) −α/(4α+1) , for α > 3/2 and for α > 1 together with the additional assumption p = o(n 4α−1 ), respectively. We deduce rate asymptotic minimax results for testing the inverse of the covariance matrix.We construct an adaptive test procedure with respect to the parameter α and show that it attains the rate ψ = (n 2 p/ ln ln(n √ p)) −α/(4α+1) .Mathematics Subject Classifications 2000: 62G10, 62H15, 62G20

We observe a sample of n independent p-dimensional Gaussian vectors with Toeplitz covariance matrix Σ = [σ |i−j| ] 1≤i,j≤p and σ 0 = 1. We consider the problem of testing the hypothesis that Σ is the identity matrix asymptotically when n → ∞ and p → ∞.We suppose that the covariances σ k decrease either polynomially ( k≥1 k 2α σ 2 k ≤ L for α > 1/4 and L > 0) or exponentially ( k≥1 e 2Ak σ 2 k ≤ 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 → +∞ 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 2α−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 α > 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.

We observe n independent p−dimensional Gaussian vectors with missing coordinates, that is each value (which is assumed standardized) is observed with probability a > 0.We investigate the problem of minimax nonparametric testing that the high-dimensional covariance matrix Σ of the underlying Gaussian distribution is the identity matrix, using these partially observed vectors. Here, n and p tend to infinity and a > 0 tends to 0, asymptotically.We assume that Σ belongs to a Sobolev-type ellipsoid with parameter α > 0. When α is known, we give asymptotically minimax consistent test procedure and find the minimax separation ratesφ n,p = (a 2 n √ p) − 2α 4α+1 , under some additional constraints on n, p and a. We show that, in the particular case of Toeplitz covariance matrices,the minimax separation rates are faster,φ n,p = (a 2 np) − 2α 4α+1 . We note how the "missingness" parameter a deteriorates the rates with respect to the case of fully observed vectors (a = 1).We also propose adaptive test procedures, that is free of the parameter α in some interval, and show that the loss of rate is (ln ln(a 2 n √ p)) α/(4α+1) and (ln ln(a 2 np)) α/(4α+1)for Toeplitz covariance matrices, respectively.Mathematics Subject Classifications 2000: 62G10, 62H15

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