We prove a general multidimensional invariance principle for a family of
U-statistics based on freely independent non-commutative random variables of
the type Un​(S), where Un​(x) is the n-th Chebyshev polynomial and S is
a standard semicircular element on a fixed W∗-probability space. As a
consequence, we deduce that homogeneous sums based on random variables of this
type are universal with respect to both semicircular and free Poisson
approximations.
Our results are stated in a general multidimensional setting and can be seen
as a genuine extension of some recent findings by Deya and Nourdin; our
techniques are based on the combination of the free Lindeberg method and the
Fourth moment Theorem