  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, lower-block-triangular and block-diagonal pseudomatrices are
asymptotically free, monotone independent and boolean independent,
respectively.
