Last month, I worked on a literature review project on "Random matrices and Wigner's Semi-circle law" as part of my course. During the project I happened to encounter a new concept called "free random variables" and the associated theory of "Free probability" as being a non-commutative analogue of the Classical Probability. I found the topic to be quite interesting from the little I understood about it (I read the introduction given in Terence Tao's book Topics in Random Matrix Theory).
So to understand more, I tried to read Dr. Roland Speicher's notes on Free probability but couldn't comprehend the Math involved. I have not yet been exposed to the topics that were used in those notes.
Hence I am putting this question here. I want to learn more about free probability and am looking for a text that talks in the context of random variables and not operator algebras (those notes involved this topic, but I do not know anything about operator algebras. By the context of random variables, I mean the way Free probability is introduced in Tao's book as something that discards information about probability spaces and concerns with the space of random variables alone). I don't have much knowledge about the subject, so I don't know if my request makes sense or not. So it will also be helpful if you can guide me about how should I approach this subject and what necessary background I will require to learn it.
(I am a Master's student in Statistics in India. I have undertaken semester courses in measure theoretic probability, real analysis and linear algebra. And I am comfortable with these topics.)
P.S. There are no existing tags on Free probability and I cannot create one. So I am putting a tag of Random matrices because it being the area where I encountered this topic.