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135052a | 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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import Mathlib
import OAI.Probability.SKBarriers.Parisi.CDFTimeModulus
namespace OAI
section
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
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
end OAI
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