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Registry leaf card · probability.empirical-covariance.unit-rank-one-moments

unitRankOneCovariance_momentPackage

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- Expectation-level covariance identities for a unit-vector random variable. If `X` takes values on the unit sphere and `C = ∫ ω, X(ω) X(ω)ᵀ ∂μ`, then this theorem proves together that 1. `C` is self-adjoint; 2. `‖C‖ ≤ 1`; 3. `∫ (X Xᵀ - C)² = C - C²`; 4. `‖C - C²‖ ≤ 2‖C‖`. This is the complete model-independent moment calculation used before a self-adjoint matrix Bernstein theorem. Independence, a matrix moment-generating function estimate, and every probability tail bound remain outside this result.

Plain-English statement

- Expectation-level covariance identities for a unit-vector random variable. If `X` takes values on the unit sphere and `C = ∫ ω, X(ω) X(ω)ᵀ ∂μ`, then this theorem proves together that 1. `C` is self-adjoint; 2. `‖C‖ ≤ 1`; 3. `∫ (X Xᵀ - C)² = C - C²`; 4. `‖C - C²‖ ≤ 2‖C‖`. This is the complete model-independent moment calculation used before a self-adjoint matrix Bernstein theorem. Independence, a matrix moment-generating function estimate, and every probability tail bound remain outside this result.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.
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Lean statement

theorem unitRankOneCovariance_momentPackage
    [CompleteSpace E]
    {Omega : Type*} [MeasurableSpace Omega]
    {mu : Measure Omega} [IsProbabilityMeasure mu]
    (sample : Omega → E) (population : E →L[ℝ] E)
    (hsampleUnit : ∀ omega, ‖sample omega‖ = 1)
    (hRankOneIntegrable :
      Integrable (fun omega =>
        InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) mu)
    (hPopulation :
      ∫ omega, InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu =
        population) :
    IsSelfAdjoint population ∧
      ‖population‖ ≤ 1 ∧
      (∫ omega,
          (InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
            population) ^ (2 : ℕ) ∂mu =
        population - population ^ (2 : ℕ)) ∧
      ‖population - population ^ (2 : ℕ)‖ ≤ 2 * ‖population‖ := by
  let A := E →L[ℝ] E
  have hSelfAdjoint : IsSelfAdjoint population := by
    rw [ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric]
    intro x y
    have hx : Integrable
        (fun omega =>
          (InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) x) mu :=
      (ContinuousLinearMap.apply ℝ E x).integrable_comp hRankOneIntegrable
    have hy : Integrable
        (fun omega =>
          (InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) y) mu :=
      (ContinuousLinearMap.apply ℝ E y).integrable_comp hRankOneIntegrable
    rw [← hPopulation]
    calc
      inner ℝ ((∫ omega,
          InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) x) y =
          inner ℝ (∫ omega,
            (InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) x ∂mu) y := by
        rw [ContinuousLinearMap.integral_apply hRankOneIntegrable]
      _ = inner ℝ y (∫ omega,
            (InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) x ∂mu) :=
        real_inner_comm _ _
      _ = ∫ omega,
            inner ℝ y
              ((InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) x) ∂mu :=
        (integral_inner hx y).symm
      _ = ∫ omega,
            inner ℝ x
              ((InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) y) ∂mu := by
        apply integral_congr_ae
        filter_upwards with omega
        calc
          inner ℝ y
              ((InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) x) =
              inner ℝ
                ((InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) x) y :=
            real_inner_comm _ _
          _ = inner ℝ x
                ((InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) y) :=
            (InnerProductSpace.isSymmetric_rankOne_self
              (𝕜 := ℝ) (sample omega)) x y
      _ = inner ℝ x (∫ omega,
            (InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) y ∂mu) :=
        integral_inner hy x
      _ = inner ℝ x ((∫ omega,
          InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) y) := by
        rw [ContinuousLinearMap.integral_apply hRankOneIntegrable]
  have hNorm : ‖population‖ ≤ 1 := by
    rw [← hPopulation]
    calc
      ‖∫ omega,
          InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu‖ ≤
          1 * mu.real Set.univ := by
        apply norm_integral_le_of_norm_le_const
        filter_upwards with omega
        rw [norm_rankOne_self_eq_sq, hsampleUnit omega]
        norm_num
      _ = 1 := by simp
  have hright : Integrable
      (fun omega =>
        InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population) mu :=
    ((ContinuousLinearMap.mul ℝ A).flip population).integrable_comp
      hRankOneIntegrable
  have hleft : Integrable
      (fun omega =>
        population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) mu :=
    (ContinuousLinearMap.mul ℝ A population).integrable_comp
      hRankOneIntegrable
  have hconst : Integrable (fun _ : Omega => population ^ (2 : ℕ)) mu :=
    integrable_const _
  have hSecondMoment :
      ∫ omega,
          (InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
            population) ^ (2 : ℕ) ∂mu =
        population - population ^ (2 : ℕ) := by
    calc
      (∫ omega,
          (InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
            population) ^ (2 : ℕ) ∂mu) =
          ∫ omega,
            InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
              InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population -
              population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) +
              population ^ (2 : ℕ) ∂mu := by
        apply integral_congr_ae
        filter_upwards with omega
        have hp := rankOne_self_sq_of_norm_one (sample omega) (hsampleUnit omega)
        calc
          (InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
              population) ^ (2 : ℕ) =
              (InnerProductSpace.rankOne ℝ (sample omega) (sample omega)) ^ (2 : ℕ) -
                InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population -
                population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) +
                population ^ (2 : ℕ) := by
            noncomm_ring
          _ = _ := by rw [hp]
      _ = (∫ omega,
            InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) -
            (∫ omega,
              InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population ∂mu) -
            (∫ omega,
              population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) +
            (∫ _ : Omega, population ^ (2 : ℕ) ∂mu) := by
        have hthree :
            (∫ omega,
              InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
                InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population -
                population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) =
              (∫ omega,
                InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) -
              (∫ omega,
                InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population ∂mu) -
              (∫ omega,
                population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) := by
          have houter := integral_sub (hRankOneIntegrable.sub hright) hleft
          have hinner := integral_sub hRankOneIntegrable hright
          calc
            (∫ omega,
                InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
                  InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population -
                  population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) =
                (∫ omega,
                  InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
                    InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population ∂mu) -
                (∫ omega,
                  population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) := by
              simpa only [Pi.sub_apply] using houter
            _ = _ := by rw [hinner]
        have hadd := integral_add
          (hRankOneIntegrable.sub hright |>.sub hleft) hconst
        calc
          (∫ omega,
              InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
                InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population -
                population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) +
                population ^ (2 : ℕ) ∂mu) =
              (∫ omega,
                InnerProductSpace.rankOne ℝ (sample omega) (sample omega) -
                  InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population -
                  population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) +
              (∫ _ : Omega, population ^ (2 : ℕ) ∂mu) := by
            simpa only [Pi.add_apply, Pi.sub_apply] using hadd
          _ = _ := by rw [hthree]
      _ = population - population ^ (2 : ℕ) := by
        have hrightInt :
            (∫ omega,
              InnerProductSpace.rankOne ℝ (sample omega) (sample omega) * population ∂mu) =
              (∫ omega,
                InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) *
                population := by
          simpa [A, mul_apply_eq_comp] using
            ContinuousLinearMap.integral_comp_comm
              ((ContinuousLinearMap.mul ℝ A).flip population)
              hRankOneIntegrable
        have hleftInt :
            (∫ omega,
              population * InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) =
              population *
                (∫ omega,
                  InnerProductSpace.rankOne ℝ (sample omega) (sample omega) ∂mu) := by
          simpa [A, mul_apply_eq_comp] using
            ContinuousLinearMap.integral_comp_comm
              (ContinuousLinearMap.mul ℝ A population) hRankOneIntegrable
        rw [hPopulation, hrightInt, hleftInt]
        simp [hPopulation]
        noncomm_ring
  have hVariance :
      ‖population - population ^ (2 : ℕ)‖ ≤ 2 * ‖population‖ := by
    calc
      ‖population - population ^ (2 : ℕ)‖ ≤
          ‖population‖ + ‖population ^ (2 : ℕ)‖ := norm_sub_le _ _
      _ ≤ ‖population‖ + ‖population‖ ^ (2 : ℕ) := by
        gcongr
        exact norm_pow_le' _ (by norm_num)
      _ ≤ 2 * ‖population‖ := by
        nlinarith [norm_nonneg population]
  exact ⟨hSelfAdjoint, hNorm, hSecondMoment, hVariance⟩

end EmpiricalCovariance
end Probability
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/Probability/EmpiricalCovariance.lean:120published source at ed709a7f320d

Proof architecture

derive self-adjointness, contraction, the exact C-C^2 centered second moment, and its norm bound before invoking self-adjoint matrix Bernstein

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