(1)Exercice given at the end of Leure

Exercice given at the end of Leure.

Recall µ is offspring diribution on Z₊, that W_n =X_+· · ·+X_n is a random walk on Z with P(X_ =i) = µ(i+) for i ≥ −, and if F is a fore u_(F), . . . , u|F|(F) are its vertices ordered in lexicographical order.

Fix an integer j ≥ and let F :Zⁿ →R₊ be a funion invariant under cyclic shifts (meaning thatF(x) =F(x⁽ⁱ⁾) for every x∈Zⁿ andi ∈Z/nZ), and letF be a foreofj independentGW_µ random trees. Show that:

E hF

k_u(F)−, ku_(F)−, . . . , ku_n−(F)−

1^{{|F |}=n}

i= j nE

hF(X_, . . . , X_n)1^{W_n=−j}

i.

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