Concise formulae for the cumulant matrices of a random vector

Hanany Ould-Baba (1) , Vincent Robin (1) , Jérôme Antoni (2, 3)

1 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
2 LVA - Laboratoire Vibrations Acoustique
3 CeLyA - centre Lyonnais d'Acoustique

Concise formulae are given for the cumulant matrices of a real-valued (zero-mean) random vector up to order 6. In addition to usual matrix operations, they involve only the Kronecker product, the vec operator, and the commutation matrix. Orders 5 and 6 are provided here for the first time; the same method as provided in the paper can be applied to compute higher orders. An immediate consequence of these formulae is to return 1) upper bounds on the rank of the cumulant matrices and 2) the expression of the sixth-order moment matrix of a Gaussian vector. Due to their conciseness, the proposed formulae also have a computational advantage as compared to the repeated use of Leonov and Shiryaev formula.

Mots clés: Moments; Cumulants; Cumulant matrix; Commutation matrix; Independent component analysis; Rank of cumulant matrices; Moment matrix of a Gaussian vector; Moments

Domaines: Mathématiques [math]

Hanany Ould-Baba, Vincent Robin, Jérôme Antoni. Concise formulae for the cumulant matrices of a random vector. Linear Algebra and its Applications, 2015, LINEAR ALGEBRA AND ITS APPLICATIONS, 485, pp.392-416. ⟨10.1016/j.laa.2015.07.027⟩. ⟨hal-01285381⟩