Whitening long range dependence in large sample covariance matrices of multivariate stationary processes
Let π be an NΓ T data matrix which can be represented as π=π_N^1/2ππ_T^1/2 with π an NΓ T random matrix whose rows are spherically symmetric, π_T a deterministic TΓ T positive definite Toeplitz matrix, and π_N a deterministic NΓ N nonnegative definite matrix. In particular, π can have i.i.d standard Gaussian entries. We prove the weak consistency of an unbiased estimator π_T=(rΜ_i-j) of ΞΎ_Nπ_T where ΞΎ_N=N^-1tr π_N, rΜ_k is the average of the entries on the kth diagonal of T^-1π^*π. When each row of π are long range dependent, i.e. the spectral density of Toeplitz matrix π_T is regularly varying at 0 with exponent aβ (-1,0), we prove that although πΜ_T may not be consistent in spectral norm, a weaker consistency of the form π_T^-1/2πΜ_T π_T^-1/2 - ΞΎ_N πa.s.0 still holds when N,Tββ with Nβ«log^3/2 T. We also establish useful probability bounds for deviations of the above convergence. It is shown next that this is strong enough for the implementation of a whitening procedure. We then apply the above result to a complex Gaussian signal detection problem where π_N is a finite rank perturbation of the identity.
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