The Free Central Limit Theorem and Free Cumulants

Abstract

Recall from Chapter 1 that if \((\mathcal{A},\varphi )\) is a non-commutative probability space and \(\mathcal{A}_{1},\ldots,\mathcal{A}_{s}\) are subalgebras of \(\mathcal{A}\) which are free with respect to φ, then freeness gives us in principle a rule by which we can evaluate φ(a1a2⋯a k ) for any alternating word in random variables a1, a2, …, a k . Thus we can in principle calculate all mixed moments for a system of free random variables. However, we do not yet have any concrete idea of the structure of this factorization rule. This situation will be greatly clarified by the introduction of free cumulants. Classical cumulants appeared in Chapter 1, where we saw that they are intimately connected with the combinatorial notion of set partitions. Our free cumulants will be linked in a similar way to the lattice of non-crossing set partitions; the latter were introduced in combinatorics by Kreweras [113]. We will motivate the appearance of free cumulants and non-crossing partition lattices in free probability theory by examining in detail a proof of the central limit theorem by the method of moments.