Orthogonality of a Partial Sum and the Next Difference

Proved

LevyStochCalc.Probability.integral_sum_mul_of_condExp_eq_zero_readable

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

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

This theorem states that a partial sum of martingale differences is orthogonal to the next difference.

Let $P$ be a probability measure on $\Omega$ , let $(\mathcal G_k)_{k\in\mathbb N}$ be an increasing family of measurable spaces dominated by the ambient measurable space, and let $Y_k:\Omega\to\mathbb R$ . Assume:

every $Y_k$ belongs to $L^2(P)$ ;
every $Y_k$ is strongly measurable in the ambient space;
$\mathbb E[Y_k\mid\mathcal G_k]=0$ almost everywhere.

Then, for every $n\in\mathbb N$ ,

\int\left(\sum_{i<n}Y_i\right)Y_n\,dP=0.

The conditional expectation is the Lean definition condExp; the result is a general martingale-difference orthogonality tool.

Original Lean declaration header

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

theorem integral_sum_mul_of_condExp_eq_zero (n : ℕ) :
    ∫ ω, (∑ i ∈ Finset.range n, Y i ω) * Y n ω ∂P = 0

Module navigation

Module: LevyStochCalc/Probability/MartingaleDifference.

Next: integral_sq_sum_of_condExp_eq_zero.

Source excerpt, lines 52–53 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Probability_MartingaleDifference

/-
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/Probability/MartingaleDifference.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory
variable {Ω : Type*} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
variable (𝒢 : ℕ → MeasurableSpace Ω) (h𝒢le : ∀ k, 𝒢 k ≤ mΩ) (h𝒢mono : Monotone 𝒢)
  (Y : ℕ → Ω → ℝ) (hY : ∀ k, MeasureTheory.MemLp (Y k) 2 P)
  (hYmeas : ∀ k, StronglyMeasurable[𝒢 (k + 1)] (Y k))
  (hYcond : ∀ k, P[Y k | 𝒢 k] =ᵐ[P] 0)
include h𝒢le h𝒢mono hY hYmeas hYcond

Formal statement


theorem LevyStochCalc.Probability.integral_sum_mul_of_condExp_eq_zero_readable (n : ℕ) :
    ∫ ω, (∑ i ∈ Finset.range n, Y i ω) * Y n ω ∂P = 0 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Probability/MartingaleDifference.lean#L50-L72 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Probability.MartingaleDifference`, declaration `LevyStochCalc.Probability.integral_sum_mul_of_condExp_eq_zero`; 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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