The Expectation of Products of Quadratic Forms in Normal Variables

Abstract

Summary A complete explicit formula for the expectation of the product of an arbitrary number of quadratic forms in normally distributed variables is derived, extending and confirming recent results of Magnus [4]. Incidentally, some results on traces of matrix products are presented.

From moments of sum to moments of product

“…However, even in this worst case scenario, it only requires computing 524,288 terms for s = 20, which is quite manageable with today's computers. 4 A small modification of Proposition 1 enables us to compute s 1 ,...,s n for z ∼ N( , ), where is not a zero vector. This is because Y = h z has a distribution of N(h , h h) and its sth moment is given by…”

“…While explicit expressions of Q s 1 ,...,s p are available when s is small (s 4), current methods are impractical for computing Q s 1 ,...,s p even for moderately large s. This is because the number of terms grows exponentially as s increases. For s = 4, we only have 17 terms but even for s = 12, there are 171,453,343 terms (see [4,12] for a method of counting the number of terms). For s = 20, there are 6.6337 × 10 17 terms, so computing something like E[(z A 1 z) 4 4 ] is simply impossible.…”

“…For s = 4, we only have 17 terms but even for s = 12, there are 171,453,343 terms (see [4,12] for a method of counting the number of terms). For s = 20, there are 6.6337 × 10 17 terms, so computing something like E[(z A 1 z) 4 4 ] is simply impossible. Holmquist [7] provides a compact expression of Q s 1 ,...,s p but again his formula requires the construction of an n s × n s symmetric matrix, which is impractical even for fairly small n and s.…”

“…Most of the existing work express Q s 1 ,...,s p as a sum of various products of the traces of s = s 1 + · · · + s p matrices related to A i 's and (see [11][12][13]4,21,9,22] for the development of this type of formulae, see also [16] for an excellent review of quadratic forms in random variables). While explicit expressions of Q s 1 ,...,s p are available when s is small (s 4), current methods are impractical for computing Q s 1 ,...,s p even for moderately large s. This is because the number of terms grows exponentially as s increases.…”

Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

“…The product function is not only used in optimization, but in other areas as well. For instance, the product of finitely many quadratic forms in random variables has been widely studied in probability and statistics [9,10,16,18,27,28,29].…”