Third Moment Bound for a Finite Sum

Proved

LevyStochCalc.Brownian.Ito.integral_abs_sum_pow_three_le_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

This is a third absolute-moment bound for a finite family of functions.

Let $\iota$ be a finite index type, let each $f_i:\Omega\to\mathbb R$ be measurable with $|f_i|^3$ integrable, and let $c_i\in\mathbb R$ satisfy

\int_\Omega|f_i|^3\,dP\le c_i

for every $i$ . Then

\int_\Omega\left|\sum_{i\in\iota}f_i\right|^3dP \le |\iota|^2\sum_{i\in\iota}c_i.

No probability-measure assumption is required. The lemma is used to control finite sums of scalar Itô-integral increments.

Original Lean declaration header

Within namespace LevyStochCalc.Brownian.Ito. Shown in its original module context. Proof omitted.

theorem integral_abs_sum_pow_three_le {ι : Type*} [Fintype ι] {f : ι → Ω → ℝ}
    (hfm : ∀ i, Measurable (f i)) (hfi : ∀ i, Integrable (fun ω => |f i ω| ^ 3) P)
    {c : ι → ℝ} (hc : ∀ i, ∫ ω, |f i ω| ^ 3 ∂P ≤ c i) :
    ∫ ω, |∑ i, f i ω| ^ 3 ∂P ≤ (Fintype.card ι : ℝ) ^ 2 * ∑ i, c i

Module navigation

Module: LevyStochCalc/Brownian/VectorItoThirdMoment.

Supporting theorems: integrable_abs_sum_pow_three.

Next: integral_norm_pi_pow_three_le.

Source excerpt, lines 44–47 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoThirdMoment

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/VectorItoThirdMoment.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
omit [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_abs_sum_pow_three_le_readable {ι : Type*} [Fintype ι] {f : ι → Ω → ℝ}
    (hfm : ∀ i, Measurable (f i)) (hfi : ∀ i, Integrable (fun ω => |f i ω| ^ 3) P)
    {c : ι → ℝ} (hc : ∀ i, ∫ ω, |f i ω| ^ 3 ∂P ≤ c i) :
    ∫ ω, |∑ i, f i ω| ^ 3 ∂P ≤ (Fintype.card ι : ℝ) ^ 2 * ∑ i, c i := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoThirdMoment.lean#L41-L65 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoThirdMoment`, declaration `LevyStochCalc.Brownian.Ito.integral_abs_sum_pow_three_le`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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