Weak convergence for the covariance operators of a Hilbertian linear process. (Q1766074)

Let \(H\) be a separable Hilbert space, \(a_k \in L(H,H)\), and let \(\varepsilon _k\) be i.i.d.\ centred random variables with values in \(H\). The author considers the stochastic process \(X_i= \sum _{k=-\infty }^{\infty } a_k\varepsilon _{i-k}\). Under the assumptions \({\mathbf E}\| \varepsilon _0\| ^4 <\infty \) and \(\sum _{k}\| a_k\| <\infty \) he obtains the following central limit theorem for the vector of empirical covariance operators of \((X_i)\): \[ \sqrt {n}(V_{n,0},\dots ,V_{n,k})@>w>> G,\qquad n\to \infty , \] where \(V_{n,i} = \frac {1}{n} \sum _{j=1}^n \langle X_{i+j},\cdot \rangle X_j - {\mathbf E}(\langle X_i,\cdot \rangle X_0)\) are Hilbert-Schmidt operators and \(G\) is a Gaussian centred random \((n+1)\)-dimensional vector having its components in the space of Hilbert-Schmidt operators. Also, a connection to the principal component analysis of the process \((X_i)\) is mentioned.