On moments of quadratic forms in non-spherically distributed variables

Abstract

Summary. We derive the expectations of the matrices xx ^T ⊗ xx ^T and xx ^T ⊗ xx ^T ⊗ xx ^T, where ⊗ denotes a Kronecker product. From this, second and third moments of quadratic forms are obtained. The moments are derived under the following assumption on the product moments of the elements X _i of x:

For any non-negative integers α ₁, … , α ₆ with , is i) finite; ii) Zero, if any α_i is odd; and iii) invariant under permutations of the indices.

AMS 1985 subject classification::

• 62 H99

Keywords:

• Random matrices
• quadratic forms