By William McKenzie McInnes
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To illustrate condition (a) above for this example, consider p = 2, that is the second block b2 = {3, 4} of π. Then there is j1 = 3, j2 = 4 in b2 and also i1 = 1 in b1 and i2 = 6 in b3 such that j1 ∼σ i1 and j2 ∼σ i2 . Fig. 8. A circular diagram 50 4 Diagrams and multigraphs Fig. 9. A circular diagram with rows of different size (ii) Another example of a circular diagram is given in Fig. 9. It is obtained from b = [12] and π = {{1, 2, 3} , {4, 5} , {6, 7} , {8, 9} , {10, 11, 12}} σ = {{1, 4} , {2, 11} , {3, 10} , {5, 7} , {6, 8} , {9, 12}} .
3, providing a characterization of the Poisson distribution. 4 (Stein’s Lemma) Let G be a Gaussian random variable with zero mean and variance σ 2 > 0, and let W be any real-valued random variable. 25) E[W h(W )] = σ 2 × E[h (W )]. Proof. 25) is verified. 25) for every test function h as in the statement. 25) implies that E(W ) = 0, E(W 2 ) = σ 2 , and, for every n ≥ 3, E[W n+1 ] = n × σ 2 × E[W n−1 ]. 5). For the sake of completeness, we shall provide a self-contained proof of the fact that the Gaussian distribution is determined by its moments.
26), as well as some useful additional identities. The reader is referred to the already quoted references [2] and [146] for further details. 1 Incidence algebras and general Möbius inversion Let P be a finite partially ordered set with partial order . We write x ≺ y to indicate that x y and x = y. 6 Möbius functions on partially ordered sets 21 inclusion relation ⊆. In our context, P = P (b), the set of partitions of b, and is the partial order ≤ considered above. The incidence algebra associated with P , denoted by I(P ) is the collection of all those functions f : P × P → C such that f (x, y) = 0 whenever x y, endowed with the convolution operation f (x, z)g(z, y).