I have a question about a common argument in free probability theory. Often times, when we want to prove a statement about free-ness or other properties of distributions of elements in a non-commutative probability space, we pass to a random matrix model and take the large N limit. Usually in addition to (semi-)circular elements (or elements with well known distributional properties), we would like to include deterministic matrices in the argument as well.
Here is my question. I have seen proof involving statements like: "Let d be an element in non-commutative probability space $(A,\phi)$, there exists a sequence of deterministic matrices $(d_N)$ such that the limiting distribution is the distribution of d."
I'm not sure how to see the existence of such a sequence. Such a statement can be found for example in Voiculescu's old article "Circular and semicircular systems and Free Product Factors" proposition 2.3.
Thank you very much.