On the rate of convergence of the distribution of the periodogram matrix for gaussian vector processes

Let {(Xk(t))k=1, …, p}t=… -1,0,1, … be a stationary Gaussian vector process with Hermitian positive definite spectral density matrix f(λ). We assume, that for some αwith 0<α≦1 and for j, k = 1, …, p. It is proved, that the distribution of the periodogram matrix In(λ) converges with the rate 0(1/nz) to a real (in case of λ= 0, ±π) or complex (in case of λ ≠ 0, ±π) WISHART distribution with one degree of freedom and with the parameter matrix f(λ). The constant in the estimation of the rate of convergence is given. Moreover, the joint distribution of In(λ1), …, In(λs) is investigated.