By Roland Speicher

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**Extra info for Combinatorics of free probability theory [Lecture notes]**

**Sample text**

An ] π∈N C(n) π∨1n =1n = kπ [a1 , . . , an ] π∈N C(n) = ϕ(a1 · · · an ), which is true since k1 = ϕ. Let us now make the induction hypothesis that for an integer m ≥ 1 the theorem is true for all m ≤ m. 62 4. FUNDAMENTAL PROPERTIES OF FREE CUMULANTS We want to show that it also holds for m + 1. This means that for τ ∈ N C(m + 1), a sequence 1 ≤ i1 < i2 < · · · < im+1 =: n, and random variables a1 , . . , an we have to prove the validity of the following equation: kτ [A1 , . . , Am+1 ] = kτ [a1 · · · ai1 , .

N − q + 1) ≈ N q for π = {V1 , . . , Vq }. Thus lim ϕ ( N →∞ a1 + · · · + aN n √ ) = lim N →∞ N π N q−(n/2) kπ = lim N →∞ #{(r(1), . . , partitions where each block Vm consists of exactly two elements – contribute. In particular, since there are no pair partitions for n odd, we see that the odd moments vanish in the limit: a1 + · · · + aN n √ ) =0 for n odd. lim ϕ ( N →∞ N Let now n = 2k be even and consider a pair partition π = {V1 , . . , Vk }. Let (r(1), . . , r(n)) be an index-tuple corresponding to this π, (r(1), .

A(i(m)) a(i(m+1)) . . ar(n) ) r r (i(1)) (i(n)) = ϕ(ar(1) . . a(i(m−1)) a(i(m+2)) . . ar(n) )ϕ(a(i(m)) a(i(m+1)) ) r r r r (i(1)) (i(m−1)) (i(m+2)) (i(n)) = ϕ(ar(1) . . ar(m−1) ar(m+2) . . ar(n) )ci(m)i(m+1) , (i) (j) (i) where cij := ϕ(ar ar ) is the covariance of (ar )i∈I . Iterating of this will lead to the final result that kπ [i(1), . . , i(n)] is given by the product of covariances (p,q)∈π ci(p)i(q) (one factor for each block (p, q) of π). 8. Let (AN , ϕN ) (N ∈ N) and (A, ϕ) be probability spaces.