| import OAI.Probability.FreeIsing.Energy.DiscrepancyEnergy |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter Set |
| open scoped Topology ENNReal |
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| namespace Problem367.DiscrepancyEnergy |
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| theorem summable_sq_sub {ι : Type*} {c c' : ι → ℝ} |
| (hc : Summable (fun i => (c i) ^ 2)) |
| (hc' : Summable (fun i => (c' i) ^ 2)) : |
| Summable (fun i => (c i - c' i) ^ 2) := by |
| apply Summable.of_nonneg_of_le (fun _ => sq_nonneg _) _ ((hc.add hc').mul_left 2) |
| intro i |
| nlinarith [sq_nonneg (c i + c' i)] |
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| theorem tsum_sq_sub_le {ι : Type*} {c c' : ι → ℝ} |
| (hc : Summable (fun i => (c i) ^ 2)) |
| (hc' : Summable (fun i => (c' i) ^ 2)) : |
| (∑' i, (c i - c' i) ^ 2) ≤ 2 * ((∑' i, (c i) ^ 2) + ∑' i, (c' i) ^ 2) := by |
| calc |
| (∑' i, (c i - c' i) ^ 2) ≤ ∑' i, 2 * ((c i) ^ 2 + (c' i) ^ 2) := by |
| exact Summable.tsum_le_tsum (fun i => by nlinarith [sq_nonneg (c i + c' i)]) |
| (summable_sq_sub hc hc') ((hc.add hc').mul_left 2) |
| _ = _ := by rw [tsum_mul_left, hc.tsum_add hc'] |
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| theorem abs_noise_add_drift_le |
| {R X m m' A A' H : ℝ} (hR : 0 ≤ R) (hA : 0 ≤ A) (hA' : 0 ≤ A') |
| (hnoise : R ≤ 2 * (A + A')) (hX : |X| ≤ H) |
| (hm : |m| ≤ 1) (hm' : |m'| ≤ 1) : |
| |R + 2 * X * (m * A - m' * A')| ≤ (2 + 2 * H) * (A + A') := by |
| have hH : 0 ≤ H := (abs_nonneg X).trans hX |
| have hd : |m * A - m' * A'| ≤ A + A' := by |
| calc |
| |m * A - m' * A'| ≤ |m * A| + |m' * A'| := abs_sub _ _ |
| _ = |m| * A + |m'| * A' := by rw [abs_mul, abs_mul, abs_of_nonneg hA, |
| abs_of_nonneg hA'] |
| _ ≤ A + A' := by nlinarith |
| calc |
| |R + 2 * X * (m * A - m' * A')| ≤ |R| + |2 * X * (m * A - m' * A')| := |
| abs_add_le _ _ |
| _ = R + 2 * |X| * |m * A - m' * A'| := by |
| rw [abs_of_nonneg hR, abs_mul, abs_mul] |
| norm_num |
| _ ≤ R + 2 * H * (A + A') := by gcongr |
| _ ≤ (2 + 2 * H) * (A + A') := by nlinarith |
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| theorem integrable_sq_difference_row |
| {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] {μ : Measure Ω} |
| {c c' : ι → Ω → ℝ} |
| (hc : ∀ i, AEStronglyMeasurable (c i) μ) |
| (hc' : ∀ i, AEStronglyMeasurable (c' i) μ) |
| (hA : Integrable (fun ω => ∑' i, (c i ω) ^ 2) μ) |
| (hA' : Integrable (fun ω => ∑' i, (c' i ω) ^ 2) μ) |
| (hsum : ∀ᵐ ω ∂μ, Summable (fun i => (c i ω) ^ 2) ∧ |
| Summable (fun i => (c' i ω) ^ 2)) : |
| Integrable (fun ω => ∑' i, (c i ω - c' i ω) ^ 2) μ := by |
| apply ((hA.add hA').const_mul 2).mono' |
| (AEStronglyMeasurable.tsum (fun i => ((hc i).sub (hc' i)).pow 2)) |
| filter_upwards [hsum] with ω hω |
| rw [Real.norm_eq_abs, abs_of_nonneg (tsum_nonneg (fun _ => sq_nonneg _))] |
| exact tsum_sq_sub_le hω.1 hω.2 |
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| theorem integrable_noise_add_drift |
| {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] {μ : Measure Ω} |
| {c c' : ι → Ω → ℝ} {X m m' : Ω → ℝ} {H : ℝ} |
| (hc : ∀ i, AEStronglyMeasurable (c i) μ) |
| (hc' : ∀ i, AEStronglyMeasurable (c' i) μ) |
| (hXm : AEStronglyMeasurable X μ) |
| (hm : AEStronglyMeasurable m μ) (hm' : AEStronglyMeasurable m' μ) |
| (hA : Integrable (fun ω => ∑' i, (c i ω) ^ 2) μ) |
| (hA' : Integrable (fun ω => ∑' i, (c' i ω) ^ 2) μ) |
| (hsum : ∀ᵐ ω ∂μ, Summable (fun i => (c i ω) ^ 2) ∧ |
| Summable (fun i => (c' i ω) ^ 2)) |
| (hbound : ∀ᵐ ω ∂μ, |X ω| ≤ H ∧ |m ω| ≤ 1 ∧ |m' ω| ≤ 1) : |
| Integrable (fun ω => (∑' i, (c i ω - c' i ω) ^ 2) + |
| 2 * X ω * (m ω * (∑' i, (c i ω) ^ 2) - m' ω * (∑' i, (c' i ω) ^ 2))) μ := by |
| have hn := integrable_sq_difference_row hc hc' hA hA' hsum |
| have hmeas := hn.aestronglyMeasurable.add |
| ((hXm.const_mul 2).mul |
| ((hm.mul hA.aestronglyMeasurable).sub (hm'.mul hA'.aestronglyMeasurable))) |
| apply ((hA.add hA').const_mul (2 + 2 * H)).mono' hmeas |
| filter_upwards [hsum, hbound] with ω hsumω hb |
| rw [Real.norm_eq_abs] |
| exact abs_noise_add_drift_le (tsum_nonneg (fun _ => sq_nonneg _)) |
| (tsum_nonneg (fun _ => sq_nonneg _)) (tsum_nonneg (fun _ => sq_nonneg _)) |
| (tsum_sq_sub_le hsumω.1 hsumω.2) hb.1 hb.2.1 hb.2.2 |
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| theorem hasSum_integral_sq_row |
| {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] {μ : Measure Ω} |
| {c : ι → Ω → ℝ} (hc : ∀ i, AEStronglyMeasurable (c i) μ) |
| (hsum : ∀ᵐ ω ∂μ, Summable (fun i => (c i ω) ^ 2)) |
| (hA : Integrable (fun ω => ∑' i, (c i ω) ^ 2) μ) : |
| HasSum (fun i => ∫ ω, (c i ω) ^ 2 ∂μ) (∫ ω, (∑' i, (c i ω) ^ 2) ∂μ) := by |
| refine hasSum_integral_of_dominated_convergence (fun i ω => (c i ω) ^ 2) |
| (fun i => (hc i).pow 2) ?_ hsum hA ?_ |
| · intro i |
| filter_upwards with ω |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| · filter_upwards [hsum] with ω hω |
| exact hω.hasSum |
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| theorem integral_noise_add_drift |
| {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] {μ : Measure Ω} |
| {c c' : ι → Ω → ℝ} {X m m' : Ω → ℝ} {H : ℝ} |
| (hc : ∀ i, AEStronglyMeasurable (c i) μ) |
| (hc' : ∀ i, AEStronglyMeasurable (c' i) μ) |
| (hXm : AEStronglyMeasurable X μ) |
| (hm : AEStronglyMeasurable m μ) (hm' : AEStronglyMeasurable m' μ) |
| (hA : Integrable (fun ω => ∑' i, (c i ω) ^ 2) μ) |
| (hA' : Integrable (fun ω => ∑' i, (c' i ω) ^ 2) μ) |
| (hsum : ∀ᵐ ω ∂μ, Summable (fun i => (c i ω) ^ 2) ∧ |
| Summable (fun i => (c' i ω) ^ 2)) |
| (hbound : ∀ᵐ ω ∂μ, |X ω| ≤ H ∧ |m ω| ≤ 1 ∧ |m' ω| ≤ 1) : |
| (∫ ω, (∑' i, (c i ω - c' i ω) ^ 2) + |
| 2 * X ω * (m ω * (∑' i, (c i ω) ^ 2) - m' ω * (∑' i, (c' i ω) ^ 2)) ∂μ) = |
| (∑' i, ∫ ω, (c i ω - c' i ω) ^ 2 ∂μ) + |
| 2 * ∫ ω, X ω * (m ω * (∑' i, (c i ω) ^ 2) - |
| m' ω * (∑' i, (c' i ω) ^ 2)) ∂μ := by |
| have hn := integrable_sq_difference_row hc hc' hA hA' hsum |
| have hE := integrable_noise_add_drift hc hc' hXm hm hm' hA hA' hsum hbound |
| have hd : Integrable (fun ω => 2 * X ω * |
| (m ω * (∑' i, (c i ω) ^ 2) - m' ω * (∑' i, (c' i ω) ^ 2))) μ := by |
| apply (hE.sub hn).congr |
| filter_upwards with ω |
| simp only [Pi.sub_apply] |
| ring |
| have hs := hasSum_integral_sq_row (fun i => (hc i).sub (hc' i)) |
| (hsum.mono (fun ω hω => summable_sq_sub hω.1 hω.2)) hn |
| simp only [Pi.sub_apply] at hs |
| rw [integral_add hn hd, ← hs.tsum_eq] |
| congr 1 |
| simp only [mul_assoc, integral_const_mul] |
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| theorem sq_row_integrable_of_summable_integrals |
| {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] {μ : Measure Ω} |
| {c : ι → Ω → ℝ} |
| (hc : ∀ i, Integrable (fun ω => (c i ω) ^ 2) μ) |
| (hs : Summable (fun i => ∫ ω, (c i ω) ^ 2 ∂μ)) : |
| (∀ᵐ ω ∂μ, Summable (fun i => (c i ω) ^ 2)) ∧ |
| Integrable (fun ω => ∑' i, (c i ω) ^ 2) μ := by |
| have hm (i : ι) : AEMeasurable (fun ω => ENNReal.ofReal ((c i ω) ^ 2)) μ := |
| (hc i).aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| have hfin : (∫⁻ ω, ∑' i, ENNReal.ofReal ((c i ω) ^ 2) ∂μ) < ∞ := by |
| rw [lintegral_tsum hm] |
| calc |
| (∑' i, ∫⁻ ω, ENNReal.ofReal ((c i ω) ^ 2) ∂μ) = |
| ∑' i, ENNReal.ofReal (∫ ω, (c i ω) ^ 2 ∂μ) := by |
| apply tsum_congr |
| intro i |
| exact (ofReal_integral_eq_lintegral_ofReal (hc i) |
| (Eventually.of_forall (fun ω => sq_nonneg (c i ω)))).symm |
| _ < ∞ := hs.tsum_ofReal_lt_top |
| have hAe : ∀ᵐ ω ∂μ, Summable (fun i => (c i ω) ^ 2) := by |
| filter_upwards [ae_lt_top' (AEMeasurable.tsum hm) hfin.ne] with ω hω |
| simpa only [ENNReal.toReal_ofReal (sq_nonneg _)] using |
| ENNReal.summable_toReal hω.ne |
| refine ⟨hAe, ?_⟩ |
| have hI := integrable_toReal_of_lintegral_ne_top (AEMeasurable.tsum hm) hfin.ne |
| simpa only [ENNReal.tsum_toReal_eq (fun i => ENNReal.ofReal_ne_top), |
| ENNReal.toReal_ofReal (sq_nonneg _)] using hI |
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| def density {Ω ι : Type*} (c c' : ι → Ω → ℝ) (X m m' : Ω → ℝ) (ω : Ω) : ℝ := |
| (∑' i, (c i ω - c' i ω) ^ 2) + |
| 2 * X ω * (m ω * (∑' i, (c i ω) ^ 2) - m' ω * (∑' i, (c' i ω) ^ 2)) |
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| theorem secondMoment_row_analysis_of_martingale |
| {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Countable ι] |
| {μ : Measure Ω} [IsFiniteMeasure μ] {F : Filtration ℝ mΩ} |
| {c c' : ι → ℝ → Ω → ℝ} {X M m m' : ℝ → Ω → ℝ} {H T : ℝ} |
| (hT : 0 ≤ T) (hX0 : Integrable (fun ω => (X 0 ω) ^ 2) μ) |
| (hM : Martingale M F μ) (hM0 : M 0 =ᵐ[μ] fun _ => 0) |
| (hc : ∀ i, AEStronglyMeasurable (fun p : ℝ × Ω => c i p.1 p.2) |
| ((volume.restrict (Ioc 0 T)).prod μ)) |
| (hc' : ∀ i, AEStronglyMeasurable (fun p : ℝ × Ω => c' i p.1 p.2) |
| ((volume.restrict (Ioc 0 T)).prod μ)) |
| (hXm : AEStronglyMeasurable (fun p : ℝ × Ω => X p.1 p.2) |
| ((volume.restrict (Ioc 0 T)).prod μ)) |
| (hm : AEStronglyMeasurable (fun p : ℝ × Ω => m p.1 p.2) |
| ((volume.restrict (Ioc 0 T)).prod μ)) |
| (hm' : AEStronglyMeasurable (fun p : ℝ × Ω => m' p.1 p.2) |
| ((volume.restrict (Ioc 0 T)).prod μ)) |
| (hA : Integrable (fun p : ℝ × Ω => ∑' i, (c i p.1 p.2) ^ 2) |
| ((volume.restrict (Ioc 0 T)).prod μ)) |
| (hA' : Integrable (fun p : ℝ × Ω => ∑' i, (c' i p.1 p.2) ^ 2) |
| ((volume.restrict (Ioc 0 T)).prod μ)) |
| (hsum : ∀ᵐ p ∂(volume.restrict (Ioc 0 T)).prod μ, |
| Summable (fun i => (c i p.1 p.2) ^ 2) ∧ |
| Summable (fun i => (c' i p.1 p.2) ^ 2)) |
| (hbound : ∀ᵐ p ∂(volume.restrict (Ioc 0 T)).prod μ, |
| |X p.1 p.2| ≤ H ∧ |m p.1 p.2| ≤ 1 ∧ |m' p.1 p.2| ≤ 1) |
| (hdecomp : ∀ t ∈ Icc 0 T, ∀ᵐ ω ∂μ, |
| (X t ω) ^ 2 = (X 0 ω) ^ 2 + M t ω + |
| ∫ s in (0 : ℝ)..t, density (fun i => c i s) (fun i => c' i s) |
| (X s) (m s) (m' s) ω) : |
| AbsolutelyContinuousOnInterval (fun t => ∫ ω, (X t ω) ^ 2 ∂μ) 0 T ∧ |
| (∀ᵐ t ∂volume.restrict (Icc 0 T), |
| deriv (fun s => ∫ ω, (X s ω) ^ 2 ∂μ) t = |
| (∑' i, ∫ ω, (c i t ω - c' i t ω) ^ 2 ∂μ) + |
| 2 * ∫ ω, X t ω * (m t ω * (∑' i, (c i t ω) ^ 2) - |
| m' t ω * (∑' i, (c' i t ω) ^ 2)) ∂μ) := by |
| have hE : Integrable (fun p : ℝ × Ω => |
| density (fun i => c i p.1) (fun i => c' i p.1) |
| (X p.1) (m p.1) (m' p.1) p.2) |
| ((volume.restrict (Ioc 0 T)).prod μ) := |
| integrable_noise_add_drift hc hc' hXm hm hm' hA hA' hsum hbound |
| obtain ⟨hac, hd⟩ := secondMoment_analysis_of_martingale hT hX0 hM hM0 hE hdecomp |
| refine ⟨hac, ?_⟩ |
| rw [← restrict_Ioc_eq_restrict_Icc] at hd ⊢ |
| have hcs := ae_all_iff.mpr (fun i => (hc i).prodMk_left) |
| have hcs' := ae_all_iff.mpr (fun i => (hc' i).prodMk_left) |
| filter_upwards [hd, hcs, hcs', hXm.prodMk_left, hm.prodMk_left, hm'.prodMk_left, |
| hA.prod_right_ae, hA'.prod_right_ae, Measure.ae_ae_of_ae_prod hsum, |
| Measure.ae_ae_of_ae_prod hbound] with t hdt hct hct' hXt hmt hmt' hAt hAt' hst hbt |
| exact hdt.trans (integral_noise_add_drift hct hct' hXt hmt hmt' hAt hAt' hst hbt) |
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| end Problem367.DiscrepancyEnergy |
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| end |
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| end OAI |
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