| import OAI.Probability.InvariantIsing.Gaussian.GaussianSingularFluctuation |
| import OAI.Probability.InvariantIsing.Pressure.RandomSqrtLimit |
|
|
| namespace OAI |
|
|
| /-! Scaled centered Gaussian squares are uniformly integrable using variance alone. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal |
| namespace InvariantIsing |
|
|
| lemma finiteGaussian_centeredSquare_integrable {ι : Type*} [Fintype ι] |
| {f : EuclideanSpace ℝ ι → ℝ} (hf : LipschitzWith 1 f) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (Z : Ω → EuclideanSpace ℝ ι) (hZ : HasLaw Z (stdGaussian _) P) (r : ℝ) : |
| Integrable (fun ω => (f (Z ω)-(∫ x, f x ∂stdGaussian _))^2/r) P := by |
| have h := (hZ.memLp_comp (finiteGaussian_lipschitz_memLp_two hf)).sub |
| (memLp_const (∫ x, f x ∂stdGaussian _)) |
| exact h.integrable_sq.div_const r |
|
|
| lemma finiteGaussian_centeredSquare_L1_bound {ι : Type*} [Fintype ι] |
| {f : EuclideanSpace ℝ ι → ℝ} (hf : LipschitzWith 1 f) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (Z : Ω → EuclideanSpace ℝ ι) (hZ : HasLaw Z (stdGaussian _) P) |
| (r : ℝ) (hr : 0 < r) : |
| eLpNorm (fun ω => (f (Z ω)-(∫ x, f x ∂stdGaussian _))^2/r) 1 P ≤ |
| ENNReal.ofReal (1/r) := by |
| have hi := finiteGaussian_centeredSquare_integrable hf P Z hZ r |
| rw [eLpNorm_one_eq_lintegral_enorm hi.aestronglyMeasurable, |
| ← ofReal_integral_norm_eq_lintegral_enorm hi] |
| have he : (∫ ω, ‖(f (Z ω)-(∫ x, f x ∂stdGaussian _))^2/r‖ ∂P) = |
| variance f (stdGaussian _)/r := by |
| have hn (ω) : ‖(f (Z ω)-(∫ x, f x ∂stdGaussian _))^2/r‖ = |
| (f (Z ω)-(∫ x, f x ∂stdGaussian _))^2/r := |
| Real.norm_of_nonneg (div_nonneg (sq_nonneg _) hr.le) |
| simp_rw [hn] |
| rw [integral_div] |
| have h := hZ.integral_comp (f := fun y => (f y-(∫ x, f x ∂stdGaussian _))^2) |
| ((hf.continuous.sub continuous_const).pow 2).aestronglyMeasurable |
| rw [show (∫ ω, (f (Z ω)-(∫ x, f x ∂stdGaussian _))^2 ∂P) = |
| ∫ y, (f y-(∫ x, f x ∂stdGaussian _))^2 ∂stdGaussian _ from h] |
| rw [← variance_eq_integral hf.continuous.aemeasurable] |
| rw [he] |
| apply ENNReal.ofReal_le_ofReal |
| apply div_le_div_of_nonneg_right _ hr.le |
| simpa only [NNReal.coe_one,one_pow] using finiteGaussian_lipschitz_variance hf |
|
|
| theorem gaussianCenteredSquare_uniformIntegrable {ι : ℕ → Type*} [∀ k, Fintype (ι k)] |
| (f : (k : ℕ) → EuclideanSpace ℝ (ι k) → ℝ) (hf : ∀ k, LipschitzWith 1 (f k)) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (Z : (k : ℕ) → Ω → EuclideanSpace ℝ (ι k)) |
| (hZ : ∀ k, HasLaw (Z k) (stdGaussian _) P) : |
| UniformIntegrable (fun k ω => |
| (f k (Z k ω)-(∫ x, f k x ∂stdGaussian _))^2/(k+1)) 1 P := by |
| have hi k := finiteGaussian_centeredSquare_integrable (hf k) P (Z k) (hZ k) (k+1) |
| have hb k := finiteGaussian_centeredSquare_L1_bound (hf k) P (Z k) (hZ k) (k+1) |
| (by positivity) |
| have ht : Tendsto (fun k : ℕ => ENNReal.ofReal (1/(k+1 : ℝ))) atTop (𝓝 0) := by |
| have h := ENNReal.continuous_ofReal.continuousAt.tendsto.comp |
| (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) |
| simpa only [ENNReal.ofReal_zero,Function.comp_def] using h |
| refine ⟨unifIntegrable_of_tendsto_Lp_zero le_rfl (by simp) |
| (fun k => memLp_one_iff_integrable.mpr (hi k)) (tendsto_of_tendsto_of_tendsto_of_le_of_le |
| tendsto_const_nhds ht (fun _ => zero_le) hb),1,?_⟩ |
| intro k |
| apply (hb k).trans |
| apply ENNReal.ofReal_le_one.mpr |
| exact (div_le_one (by positivity : (0 : ℝ) < k+1)).mpr (by have := Nat.cast_nonneg (α := ℝ) k; linarith) |
|
|
| end InvariantIsing |
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| end |
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| end OAI |
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