Asymptotic properties of random matrices and pseudomatrices

TL;DR

This paper investigates the asymptotic behavior of random pseudomatrices and compares it with that of block-structured random matrices, revealing new forms of asymptotic independence and their operator realizations.

Contribution

It introduces the concept of matricially free Gaussian operators and characterizes the asymptotic distributions and independence properties of blocks in random pseudomatrices.

Findings

Asymptotic joint distributions of blocks are derived and realized via matricially free Gaussian operators.

Symmetric blocks of pseudomatrices match the asymptotics of symmetric random blocks.

Different block structures exhibit asymptotic freeness, monotone independence, or boolean independence.

Abstract

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.