Characterization of non-commutative free Gaussian variables
We provide a necessary and sufficient condition, based on zero correlation, for self-adjoint, freely independent, identically distributed random variables on a *-probability space to be free Gaussian. Along the way, we establish a free analogue of a well known application of Basu's theorem from statistics. We also show that all linear combinations of free Gaussian being free Gaussian does not necessarily imply joint free Gaussianity, and we identify additional conditions under which this implication is true.
Bose, Arup; Dey, Apratim; and Ejsmont, Wiktor, "Characterization of non-commutative free Gaussian variables" (2018). Journal Articles. 1528.