| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.CDFTimeModulus |
|
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| namespace OAI |
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| section |
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| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem uniform_derivative_convergence_filter {ι : Type*} {l : Filter ι} |
| {F D : ι → ℝ → ℝ} {f d : ℝ → ℝ} (K : ℝ≥0) |
| (hD : ∀ᶠ n in l, ∀ x, HasDerivAt (F n) (D n x) x) |
| (hd : ∀ x, HasDerivAt f (d x) x) |
| (hLip : ∀ᶠ n in l, LipschitzWith K (D n)) (hlip : LipschitzWith K d) |
| (hF : TendstoUniformly F f l) : TendstoUniformly D d l := by |
| rw [Metric.tendstoUniformly_iff] at hF ⊢ |
| intro ε hε |
| let h : ℝ := ε/(8*((K:ℝ)+1)) |
| have hh : 0<h := div_pos hε (by positivity) |
| have hke : 8*((K:ℝ)+1)*h=ε := by dsimp [h]; field_simp |
| let δ : ℝ := ε*h/8 |
| have hδ : 0<δ := by dsimp [δ]; positivity |
| filter_upwards [hF δ hδ,hD,hLip] with n hn hdn hLn x |
| have hb (z : ℝ) : |F n z-f z| ≤ δ := by simpa only [Real.dist_eq,abs_sub_comm] using (hn z).le |
| have H := derivative_bound_of_uniform_value_and_lipschitz _ _ (K+K) δ |
| (fun z => (hdn z).sub (hd z)) (hLn.sub hlip) hb x h hh |
| have he : 2*δ/h=ε/4 := by dsimp [δ]; field_simp; ring |
| rw [he,NNReal.coe_add] at H |
| rw [Real.dist_eq,abs_sub_comm] |
| have hk := K.coe_nonneg |
| nlinarith |
|
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| theorem scalarCDFValue_time_modulus_symmetric (β : ℝ) (α : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (h1 : α 1=1) {s q : ℝ} |
| (hs : s∈Icc (0:ℝ) 1) (hq : q∈Icc (0:ℝ) 1) (x : ℝ) : |
| |scalarCDFValue β α s (Real.toNNReal (1-s)) x- |
| scalarCDFValue β α q (Real.toNNReal (1-q)) x| ≤ |
| scalarTimeMassConstantK β 1*|s-q|+|β| *Real.sqrt |s-q| *standardGaussianAbsMoment := by |
| rcases le_total s q with h | h |
| · have H := scalarCDFValue_time_modulus β α ha h1 hs.1 h hq.2 x |
| simpa only [abs_of_nonpos (sub_nonpos.mpr h),neg_sub] using H |
| · have H := scalarCDFValue_time_modulus β α ha h1 hq.1 h hs.2 x |
| rw [abs_sub_comm] at H |
| simpa only [abs_of_nonneg (sub_nonneg.mpr h)] using H |
|
|
| theorem scalarCDFValue_time_uniform (β : ℝ) (α : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (h1 : α 1=1) {q : ℝ} (hq : q∈Icc (0:ℝ) 1) : |
| TendstoUniformly (fun s => scalarCDFValue β α s (Real.toNNReal (1-s))) |
| (scalarCDFValue β α q (Real.toNNReal (1-q))) (𝓝[Icc (0:ℝ) 1] q) := by |
| have hc : Continuous (fun s : ℝ => scalarTimeMassConstantK β 1*|s-q|+ |
| |β| *Real.sqrt |s-q| *standardGaussianAbsMoment) := by fun_prop |
| have ht : Tendsto (fun s : ℝ => scalarTimeMassConstantK β 1*|s-q|+ |
| |β| *Real.sqrt |s-q| *standardGaussianAbsMoment) (𝓝[Icc (0:ℝ) 1] q) (𝓝 0) := by |
| simpa only [sub_self,abs_zero,mul_zero,Real.sqrt_zero,zero_mul,zero_add] using |
| (hc.continuousAt (x := q)).tendsto.mono_left nhdsWithin_le_nhds |
| apply Metric.tendstoUniformly_iff.mpr |
| intro ε hε |
| filter_upwards [(tendsto_order.mp ht).2 ε hε,self_mem_nhdsWithin] with s hs hsm x |
| rw [Real.dist_eq,abs_sub_comm] |
| exact (scalarCDFValue_time_modulus_symmetric β α ha h1 hsm hq x).trans_lt hs |
|
|
| theorem scalarCDFGradient_time_uniform (β : ℝ) (α : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (h1 : α 1=1) {q : ℝ} (hq : q∈Icc (0:ℝ) 1) : |
| TendstoUniformly (fun s => scalarCDFGradient β α s (Real.toNNReal (1-s))) |
| (scalarCDFGradient β α q (Real.toNNReal (1-q))) (𝓝[Icc (0:ℝ) 1] q) := by |
| have hr {s : ℝ} (hs : s∈Icc (0:ℝ) 1) : Real.toNNReal (1-s) ≤ 1 := by |
| rw [Real.toNNReal_le_iff_le_coe,NNReal.coe_one]; linarith [hs.1] |
| apply uniform_derivative_convergence_filter 1 _ |
| (scalarCDFValue_hasDerivAt β ha α.mono q _ (hr hq)) _ |
| (scalarCDFGradient_lipschitz β ha α.mono q _ (hr hq)) |
| (scalarCDFValue_time_uniform β α ha h1 hq) |
| · filter_upwards [self_mem_nhdsWithin] with s hs |
| exact scalarCDFValue_hasDerivAt β ha α.mono s _ (hr hs) |
| · filter_upwards [self_mem_nhdsWithin] with s hs |
| exact scalarCDFGradient_lipschitz β ha α.mono s _ (hr hs) |
|
|
| theorem scalarCDFHessian_time_uniform (β : ℝ) (α : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (h1 : α 1=1) {q : ℝ} (hq : q∈Icc (0:ℝ) 1) : |
| TendstoUniformly (fun s => scalarCDFHessian β α s (Real.toNNReal (1-s))) |
| (scalarCDFHessian β α q (Real.toNNReal (1-q))) (𝓝[Icc (0:ℝ) 1] q) := by |
| have hr {s : ℝ} (hs : s∈Icc (0:ℝ) 1) : Real.toNNReal (1-s) ≤ 1 := by |
| rw [Real.toNNReal_le_iff_le_coe,NNReal.coe_one]; linarith [hs.1] |
| apply uniform_derivative_convergence_filter susceptibilityLipschitzConstant _ |
| (scalarCDFGradient_hasDerivAt β ha α.mono q _ (hr hq)) _ |
| (scalarCDFHessian_lipschitz β ha α.mono q _ (hr hq)) |
| (scalarCDFGradient_time_uniform β α ha h1 hq) |
| · filter_upwards [self_mem_nhdsWithin] with s hs |
| exact scalarCDFGradient_hasDerivAt β ha α.mono s _ (hr hs) |
| · filter_upwards [self_mem_nhdsWithin] with s hs |
| exact scalarCDFHessian_lipschitz β ha α.mono s _ (hr hs) |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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