Combinatorics of free probability theory [Lecture notes] by Roland Speicher

By Roland Speicher

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An ] − km+1 [A1 , . . , Am+1 ] = = kπ [a1 , . . , an ] π∈N C(n) π∨σ=1n π∈N C(n) kπ [a1 , . . , an ] . π∈N C(n) π∨σ=1n The second proof reduces the statement essentially to some general structure theorem on lattices. This is the well-known M¨obius algebra which we present here in a form which looks analogous to the description of convolution via Fourier transform. 5. Let P be a finite lattice. 1) For two functions f, g : P → C on P we denote by f ∗ g : P → C the function defined by (104) f ∗ g(π) := f (σ1 )g(σ2 ) σ1 ,σ2 ∈P σ1 ∨σ2 =π (π ∈ P ).

I=1 3) The same argumentation works also for a proof of the classical central limit theorem. The only difference appears when one determines the contribution of pair partitions. In contrast to the free case, all partitions give now the same factor. We will see in the forthcoming sections that this is a special case of the following general fact: the transition from classical to free probability theory consists on a combinatorial level in the transition from all partitions to non-crossing partitions.

Ar = µap for all r, p). Furthermore, assume that all variables are centered, ϕ(ar ) = 0 (r ∈ N), and denote by σ 2 := ϕ(a2r ) the common variance of the variables. Then we have a1 + · · · + aN distr √ −→ γ, N 37 38 3. FREE CUMULANTS where γ is a normally distributed random variable of variance σ 2 . Explicitly, this means (61) lim ϕ ( N →∞ a1 + · · · + aN n 1 √ ) =√ N 2πσ 2 2 /2σ 2 tn e−t dt ∀ n ∈ N. 4. (free CLT: one-dimensional case) Let (A, ϕ) be a probability space and a1 , a2 , · · · ∈ A a sequence of free and identically distributed random variables.

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