Spectral analysis of the fourth moment matrix
Let \(x = (X_1, \dots , X_d)\) be a \(d\)-dimensional random vector satisfying \(E(X_i^4)<+\infty\), for \(i = 1,\dots, d\). The fourth moment of \(x\) is the \(d^2\times d^2\) symmetric and positive semidefinite matrix \(\mu_4(x)=E(x\otimes x^T\otimes x\otimes x^T)\), where \(\otimes\) denotes the Kronecker product. In this paper, a lower bound for the fourth moment of \(x\) dominant eigenvalue is provided. It is shown that its eigenvectors corresponding to positive eigenvalues are vectorized symmetric matrices. Fourth moments of standardized and exchangeable random vectors are examined.
scientific article; zbMATH DE number 5924555