| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.CDFOverlapConvergence |
| import OAI.Probability.SKBarriers.Parisi.CDFMetric |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem uniform_real_mul_bounded {f g : ℝ → ℝ} {fn gn : ℕ → ℝ → ℝ} {S : Set ℝ} |
| (hf : TendstoUniformlyOn fn f atTop S) (hg : TendstoUniformlyOn gn g atTop S) |
| (hfn : ∀ n x, x∈S → |fn n x|≤1) {C : ℝ} (hC : 0≤C) (hgb : ∀ x, x∈S → |g x|≤C) : |
| TendstoUniformlyOn (fun n x => fn n x*gn n x) (fun x => f x*g x) atTop S := by |
| apply Metric.tendstoUniformlyOn_iff.mpr |
| intro ε hε |
| let δ := ε/(2*(C+1)) |
| have hδ : 0<δ := div_pos hε (by positivity) |
| have Hf := Metric.tendstoUniformlyOn_iff.mp hf δ hδ |
| have Hg := Metric.tendstoUniformlyOn_iff.mp hg δ hδ |
| filter_upwards [Hf,Hg] with n hnf hng x hx |
| rw [Real.dist_eq,abs_sub_comm] |
| have he : fn n x*gn n x-f x*g x=fn n x*(gn n x-g x)+(fn n x-f x)*g x := by ring |
| rw [he] |
| calc |
| _ ≤ |fn n x| * |gn n x-g x|+|fn n x-f x| * |g x| := by simpa only [abs_mul] using abs_add_le (fn n x*(gn n x-g x)) ((fn n x-f x)*g x) |
| _ ≤ 1*δ+δ*C := add_le_add |
| (mul_le_mul (hfn n x hx) (by simpa only [Real.dist_eq,abs_sub_comm] using (hng x hx).le) (abs_nonneg _) (by norm_num)) |
| (mul_le_mul (by simpa only [Real.dist_eq,abs_sub_comm] using (hnf x hx).le) (hgb x hx) (abs_nonneg _) hδ.le) |
| _ < ε := by dsimp [δ]; field_simp; nlinarith |
|
|
| theorem uniformCDFQuantiles_tendstoUniformlyOn (α : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (h1 : α 1=1) : |
| TendstoUniformlyOn (fun n => quantileCDF n (uniformCDFQuantiles n α)) α atTop (Icc (0:ℝ) 1) := by |
| apply Metric.tendstoUniformlyOn_iff.mpr |
| intro ε hε |
| have H : Tendsto (fun n : ℕ => ((n+1:ℕ):ℝ)⁻¹) atTop (𝓝 0) := by |
| simpa only [Nat.cast_add,Nat.cast_one,one_div] using (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜:=ℝ)) |
| filter_upwards [(tendsto_order.mp H).2 ε hε] with n hn x hx |
| rw [Real.dist_eq,abs_sub_comm] |
| exact (quantileCDF_uniformCDFQuantiles_error n α ha h1 hx).trans_lt hn |
|
|
| theorem scalarCDFVariation_integrable {β : ℝ} {α γ : ℝ → ℝ} |
| (hα : Monotone α) (hγ : Monotone γ) |
| (hΓ : ContinuousOn (scalarCDFOverlap β α) (Icc (0:ℝ) 1)) : |
| IntegrableOn (fun s => (γ s-α s)*(scalarCDFOverlap β α s-s)) (Icc (0:ℝ) 1) := |
| ((hγ.monotoneOn _).integrableOn_isCompact isCompact_Icc).sub |
| ((hα.monotoneOn _).integrableOn_isCompact isCompact_Icc) |>.mul_continuousOn |
| (hΓ.sub continuousOn_id) isCompact_Icc |
|
|
| theorem scalarCDFVariation_quantile_integral_tendsto (β : ℝ) (α γ : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (hα1 : α 1=1) |
| (hg : ∀ z, γ z∈Icc (0:ℝ) 1) (hγ1 : γ 1=1) : |
| Tendsto (fun n => ∫ s in Icc (0:ℝ) 1, |
| (quantileCDF n (uniformCDFQuantiles n γ) s-quantileCDF n (uniformCDFQuantiles n α) s)* |
| (scalarCDFOverlap β (quantileCDF n (uniformCDFQuantiles n α)) s-s)) atTop |
| (𝓝 (∫ s in Icc (0:ℝ) 1, (γ s-α s)*(scalarCDFOverlap β α s-s))) := by |
| have HA := uniformCDFQuantiles_tendstoUniformlyOn α ha hα1 |
| have HG := uniformCDFQuantiles_tendstoUniformlyOn γ hg hγ1 |
| have HΓ := scalarCDFOverlap_tendstoUniformlyOn β ha α.mono |
| (fun n => quantileCDF_bounds n (uniformCDFQuantiles n α)) |
| (fun n => quantileCDF_monotone n (uniformCDFQuantiles n α)) |
| (uniformCDFQuantiles_L1_tendsto α ha hα1) |
| have Hc := scalarCDFOverlap_continuousOn β α ha hα1 |
| obtain ⟨C,hC⟩ := isCompact_Icc.exists_bound_of_continuousOn (Hc.sub continuousOn_id) |
| apply unitInterval_tendsto_integral_uniform (scalarCDFVariation_integrable α.mono γ.mono Hc) |
| (fun n => scalarCDFVariation_integrable (quantileCDF_monotone n _) (quantileCDF_monotone n _) |
| (scalarCDFOverlap_continuousOn_quantile β _ (uniformCDFQuantiles_admissible n α hα1))) |
| have HId : TendstoUniformlyOn (fun _ : ℕ => fun s : ℝ => s) (fun s => s) atTop (Icc (0:ℝ) 1) := by |
| apply Metric.tendstoUniformlyOn_iff.mpr |
| intro ε hε |
| exact Eventually.of_forall (fun _ _ _ => by simpa using hε) |
| apply uniform_real_mul_bounded (HG.fun_sub HA) (HΓ.fun_sub HId) _ |
| (C:=|C|) (abs_nonneg C) _ |
| · intro n x _ |
| have H₁ := quantileCDF_bounds n (uniformCDFQuantiles n α) x |
| have H₂ := quantileCDF_bounds n (uniformCDFQuantiles n γ) x |
| exact abs_le.mpr ⟨by linarith [H₁.2,H₂.1],by linarith [H₁.1,H₂.2]⟩ |
| · intro x hx |
| have H : |scalarCDFOverlap β α x-x| ≤ C := by simpa only [Real.norm_eq_abs, Pi.sub_apply, id_eq] using hC x hx |
| exact H.trans (le_abs_self C) |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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