Remarks on Limit Theorems for the Free Quadratic Forms

In 2021, Ejsmont and Biernacki showed that the free tangent distribution can be used to measure household satisfaction with durable consumer goods. This distribution arises as the limit of free random variables. This, new article serves as the theoretical introduction to the continuation of the research presented in the paper from 2021. We continue the study of the limit of specific quadratic forms in free probability, which is the first step towards constructing a new distribution for the evaluation of satisfaction with material affluence among household. We formulate a non-central limit theorem for weighted sums of commutators and square of the sums for free random variable. In addition we give the random matrix models for these limits.