Multidimensional semicircular limits on the free Wigner chaos
Ivan Nourdin (IECN), Giovanni Peccati, Roland Speicher
TL;DR
This paper proves that for sequences of vectors of multiple Wigner integrals, componentwise convergence to semicircular distributions implies joint convergence, extending classical results to the free probability setting and multidimensional cases.
Contribution
It establishes a multidimensional semicircular limit theorem for free Wigner chaos, generalizing previous unidimensional results and linking componentwise and joint convergence.
Findings
Componentwise convergence implies joint convergence in free Wigner chaos.
Extends classical probabilistic limit theorems to free probability setting.
Provides a multidimensional counterpart to existing free Wigner chaos limit theorems.
Abstract
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).
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Taxonomy
TopicsRandom Matrices and Applications · Stochastic processes and financial applications · Point processes and geometric inequalities