For epsilon-independent variables, normalized sums converge to a universal law determined by the graphon limit of the independence graph, interpolating between Gaussian and semicircle.
Central Limit Theorem for tensor products of free variables
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abstract
We establish a central limit theorem for tensor product random variables $c_k:=a_k \otimes a_k$, where $(a_k)_{k \in \mathbb{N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables $a_k$ and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.
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Graphon-Theoretic Approach to Central Limit Theorems for $\epsilon$-Independence
For epsilon-independent variables, normalized sums converge to a universal law determined by the graphon limit of the independence graph, interpolating between Gaussian and semicircle.