Bounds for free cumulants

For φ : A → R with values in a normed commutative ring, a free cumulant of a constant word (x, …, x) along s is bounded in terms of bounds for the moments φ (x ^ k) with 0 < k ≤ #s alone. Möbius inversion over the noncrossing partitions of s gives the explicit bound ∑_{π ∈ NC(s)} ‖μ(π, 1_s)‖ ∏_{B ∈ π} M #B, which depends on x only through the moment bounds M. Families whose moments are bounded uniformly therefore have uniformly bounded cumulants, as used in the cumulant proof of the free central limit theorem.

Main results

• TauCeti.norm_freeCumulant_const_le — a free cumulant of a constant word is bounded by a constant depending only on bounds for the moments of its letter.

theorem TauCeti.norm_freeCumulant_const_le {R : Type u_1} {A : Type u_2} {α : Type u_3} [NormedCommRing R] [NormOneClass R] [Monoid A] [LinearOrder α] (φ : A → R) (x : A) {s : Finset α} {M : ℕ → ℝ} (hM : ∀ (k : ℕ), 0 < k → k ≤ s.card → ‖φ (x ^ k)‖ ≤ M k) : ‖freeCumulant φ (fun (x_1 : α) => x) s‖ ≤ ∑ π : NoncrossingPartition s, ‖(IncidenceAlgebra.mu R) π ⊤‖ * ∏ B ∈ (↑π).parts, M B.card

Uniform bound for free cumulants: if the positive-order moments of x up to #s satisfy ‖φ (x ^ k)‖ ≤ M k, then the free cumulant of the constant word (x, …, x) along s is bounded by ∑_{π ∈ NC(s)} ‖μ(π, 1_s)‖ ∏_{B ∈ π} M #B, which depends on x only through M.