| import Mathlib |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
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| 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 |
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| 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 |
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| end |
| end |
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| end |
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| end OAI |
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