  We show that the operatorial framework developed by Voiculescu for free
random variables can be extended to arrays of random variables whose
multiplication imitates matricial multiplication. The associated notion of
independence, called matricial freeness, can be viewed as a generalization of
both freeness and monotone independence. At the same time, the sums of
matricially free random variables, called random pseudomatrices, are closely
related to Gaussian random matrices. The main results presented in this paper
concern the standard and tracial central limit theorems for random
pseudomatrices and the corresponding limit distributions which can be viewed as
matricial generalizations of semicirle laws.
