File size: 2,840 Bytes
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 | import Mathlib
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem mean_ge_of_subGaussian_upper {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ] {f : Ω → ℝ} {v : NNReal}
(hv : 0 < (v:ℝ))
(hsg : HasSubgaussianMGF (fun x => f x-∫ y, f y ∂μ) v μ)
{a B : ℝ} (hB : 0 ≤ B)
(hp : Real.exp (-B) ≤ μ.real {x | a ≤ f x}) :
a-Real.sqrt (2*(v:ℝ)*B) ≤ ∫ x, f x ∂μ := by
by_cases h : a ≤ ∫ x, f x ∂μ
· linarith [Real.sqrt_nonneg (2*(v:ℝ)*B)]
have hd : 0 < a-∫ x, f x ∂μ := sub_pos.mpr (not_le.mp h)
have ht := hsg.measure_ge_le hd.le
have hevent : {x | a ≤ f x} = {x | a-(∫ y, f y ∂μ) ≤ f x-(∫ y, f y ∂μ)} := by
ext x; simp only [Set.mem_ofPred_eq]; constructor <;> intro hx <;> linarith
rw [← hevent] at ht
have he := Real.exp_le_exp.mp (hp.trans ht)
have hs : (a-(∫ y, f y ∂μ))^2 ≤ 2*(v:ℝ)*B := by
have HH := (le_div_iff₀ (by positivity : 0 < 2*(v:ℝ))).mp
(show -B ≤ -(a-(∫ y, f y ∂μ))^2/(2*(v:ℝ)) from he)
nlinarith
have hr := Real.sq_sqrt (show 0 ≤ 2*(v:ℝ)*B by positivity)
have hrpos := Real.sqrt_nonneg (2*(v:ℝ)*B)
nlinarith
theorem subGaussian_half_lower {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ] {f : Ω → ℝ} {v : NNReal}
(hf : Measurable f) (hv : 0 < (v:ℝ))
(hsg : HasSubgaussianMGF (fun x => f x-∫ y, f y ∂μ) v μ)
{a B : ℝ} (hB : 0 ≤ B)
(hp : Real.exp (-B) ≤ μ.real {x | a ≤ f x}) :
(1/2:ℝ) ≤ μ.real {x | a-Real.sqrt (2*(v:ℝ)*B)-
Real.sqrt (2*(v:ℝ)*Real.log 2) ≤ f x} := by
have hm := mean_ge_of_subGaussian_upper hv hsg hB hp
let u := Real.sqrt (2*(v:ℝ)*Real.log 2)
have hu : 0 ≤ u := Real.sqrt_nonneg _
have hu2 : u^2 = 2*(v:ℝ)*Real.log 2 :=
Real.sq_sqrt (by positivity)
have ht := hsg.neg.measure_ge_le hu
simp only [Pi.neg_apply,neg_sub] at ht
have hexp : Real.exp (-u^2/(2*(v:ℝ))) = (1/2:ℝ) := by
rw [hu2]
have hvn : (v:ℝ) ≠ 0 := hv.ne'
have he : -(2*(v:ℝ)*Real.log 2)/(2*(v:ℝ)) = -Real.log 2 := by field_simp
rw [he,Real.exp_neg,Real.exp_log (by norm_num)]
norm_num
rw [hexp] at ht
let A := {x | a-Real.sqrt (2*(v:ℝ)*B)-u ≤ f x}
have hA : MeasurableSet A := measurableSet_le measurable_const hf
have hsub : Aᶜ ⊆ {x | u ≤ (∫ y, f y ∂μ)-f x} := by
intro x hx
have hx' : f x < a-Real.sqrt (2*(v:ℝ)*B)-u := not_le.mp hx
simp only [Set.mem_ofPred_eq]
linarith
have hb := (measureReal_mono hsub).trans ht
rw [measureReal_compl hA] at hb
simp only [probReal_univ] at hb
change (1/2:ℝ) ≤ μ.real A
linarith
end SK.Analytic
end
end
end
end OAI
|