On fluctuations of traces of large matrices over a non-commutative algebra

The paper investigates the asymptotic behavior of (non-normalized) traces of certain classes of matrices with non-commutative random variables as entries. We show that, unlike in the commutative framework, the asymptotic behavior of matrices with free circular, respectively with Bernoulli distributed Boolean independent entries is described in terms of free, respectively Boolean cumulants. We also present an exemple of relation of monotone independence arising from the study of Boolean independence.