Asymptotic properties of random matrices and pseudomatrices

We study the asymptotics of sums of matricially free random variables, called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called ‘matricially free Gaussian operators’. In particular, if the variance matrices are symmetric, the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks. We also show that blocks of random pseudomatrices are ‘asymptotically matricially free’ whereas the corresponding symmetric random blocks are ‘asymptotically symmetrically matricially free’, where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization. Finally, we show that row blocks of square, block-lower-triangular and block-diagonal pseudomatrices are asymptotically free, monotone independent and boolean independent, respectively.