File size: 3,774 Bytes
a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | 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
end
end OAI
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