Multidimensional semicircular limits on the free Wigner chaos

Ivan Nourdin (1) , Giovanni Peccati (2) , Roland Speicher (3)

1 IECN - Institut Élie Cartan de Nancy
2 Unité de Recherche en Mathématiques
3 Universität des Saarlandes

We show that, for sequences of vectors of multiple Wigner integrals with respect to a free Brownian motion, componentwise convergence to semicircular is equivalent to joint convergence. This result extends to the free probability setting some findings by Peccati and Tudor (2005), and represents a multidimensional counterpart of a limit theorem inside the free Wigner chaos established by Kemp, Nourdin, Peccati and Speicher (2011).

Mots clés: Free Brownian Motion; Fourth Moment Condition; Convergence in Distribution; Multidimensional Limit Theorems; Semicircular Law; Free Probability; Wigner Chaos

Domaines: Probabilités [math.PR]; Algèbres d'opérateurs [math.OA]

Ivan Nourdin, Giovanni Peccati, Roland Speicher. Multidimensional semicircular limits on the free Wigner chaos. 2011. ⟨hal-00611237⟩