| import OAI.Analysis.FiniteFactor.Sampling.SelfMoment |
| import OAI.Analysis.FiniteFactor.Sampling.TestedSecondMoment |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| noncomputable section |
|
|
| / |
| theorem sampling_square_integrable {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| [IsFiniteMeasure μ] (f : Ω → ℝ) (C : ℝ) (hm : Measurable f) |
| (hb : ∀ x, 0 ≤ f x ∧ f x ≤ C) : Integrable (fun x => f x ^ 2) μ := by |
| apply (integrable_const (C^2)).mono' (hm.pow_const 2).aestronglyMeasurable |
| exact Eventually.of_forall (fun x => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg (f x))] |
| nlinarith [(hb x).1,(hb x).2]) |
|
|
| / |
| theorem samplingWeight_square_integrable (b : Bool) (h α : ℝ) (hh : 0 < h) (hα : 0 < α) : |
| Integrable (fun p : ℝ × ℝ => samplingWeight b h α p.1 p.2 ^ 2) |
| (unitIntervalMeasure.prod unitIntervalMeasure) := |
| sampling_square_integrable _ _ (max 1 (1/(h*α))) (samplingWeight_measurable b h α) |
| (fun p => samplingWeight_bounds b h α p.1 p.2 hh hα) |
|
|
| / |
| orientation, with the source's order of both integrations. -/ |
| theorem samplingSecondWeight_square_integral (h α : ℝ) (hh : 0 < h) (hh1 : h < 1) |
| (hα : 0 < α) (hα1 : α ≤ 1/2) : |
| (∫ s, ∫ t, samplingSecondWeight h α s t ^ 2 |
| ∂unitIntervalMeasure ∂unitIntervalMeasure) = (2/h)*Real.log (1/(2*α)) := by |
| have hi : Integrable (fun p : ℝ × ℝ => samplingSecondWeight h α p.1 p.2 ^ 2) |
| (unitIntervalMeasure.prod unitIntervalMeasure) := |
| sampling_square_integrable _ _ (max 1 (1/(h*α))) (samplingSecondWeight_measurable h α) |
| (fun p => samplingWeight_bounds false h α (Int.fract (p.1-p.2)) p.2 hh hα) |
| rw [integral_integral_swap hi] |
| calc |
| _ = ∫ t, ∫ s, samplingWeight false h α s t ^ 2 ∂unitIntervalMeasure ∂unitIntervalMeasure := by |
| apply integral_congr_ae |
| filter_upwards [] with t |
| have hm : AEStronglyMeasurable (fun s => samplingWeight false h α s t ^ 2) unitIntervalMeasure := |
| (((samplingWeight_measurable false h α).comp (measurable_id.prodMk measurable_const)).pow_const 2).aestronglyMeasurable |
| have hp := unitInterval_rotate_preserving (-t) |
| have he := integral_map hp.measurable.aemeasurable (show AEStronglyMeasurable |
| (fun s => samplingWeight false h α s t ^ 2) (unitIntervalMeasure.map (fun s => Int.fract (s+ -t))) from |
| hp.map_eq.symm ▸ hm) |
| rw [hp.map_eq] at he |
| simpa [samplingSecondWeight,sub_eq_add_neg] using he.symm |
| _ = _ := by |
| rw [← integral_integral_swap (samplingWeight_square_integrable false h α hh hα), |
| samplingWeight_square_integral h α hh hh1 hα hα1] |
|
|
| / |
| theorem samplingTestedFirstWeight_square_integral (h α : ℝ) (E G : Set ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hα1 : α ≤ 1/2) |
| (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| (∫ s, ∫ t, samplingTestedFirstWeight false h α E G s t ^ 2 |
| ∂unitIntervalMeasure ∂unitIntervalMeasure) ≤ (2/h)*Real.log (1/(2*α)) := by |
| have hb (s t : ℝ) := samplingTestedFirstWeight_bounds false h α E G s t hh hα |
| have hm := samplingTestedFirstWeight_measurable false h α E G hE hG |
| have hbig := samplingWeight_square_integrable false h α hh hα |
| have hsmall : Integrable (fun p : ℝ × ℝ => samplingTestedFirstWeight false h α E G p.1 p.2 ^ 2) |
| (unitIntervalMeasure.prod unitIntervalMeasure) := |
| sampling_square_integrable _ _ (max 1 (1/(h*α))) hm |
| (fun p => ⟨(hb p.1 p.2).1,(hb p.1 p.2).2.trans (samplingWeight_bounds false h α p.1 p.2 hh hα).2⟩) |
| rw [← samplingWeight_square_integral h α hh hh1 hα hα1, |
| ← integral_prod _ hsmall,← integral_prod _ hbig] |
| apply integral_mono_ae hsmall hbig |
| exact Eventually.of_forall (fun p => by |
| nlinarith [(hb p.1 p.2).1,(hb p.1 p.2).2, |
| (samplingWeight_bounds false h α p.1 p.2 hh hα).1]) |
|
|
| / |
| theorem samplingTestedSecondWeight_square_integral (h α : ℝ) (E G : Set ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hα1 : α ≤ 1/2) |
| (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| (∫ s, ∫ t, samplingTestedSecondWeight h α E G s t ^ 2 |
| ∂unitIntervalMeasure ∂unitIntervalMeasure) ≤ (2/h)*Real.log (1/(2*α)) := by |
| have hb (s t : ℝ) := samplingTestedSecondWeight_bounds h α E G s t hh hα |
| have hm := samplingTestedSecondWeight_measurable h α E G hE hG |
| have hbig : Integrable (fun p : ℝ × ℝ => samplingSecondWeight h α p.1 p.2 ^ 2) |
| (unitIntervalMeasure.prod unitIntervalMeasure) := |
| sampling_square_integrable _ _ (max 1 (1/(h*α))) (samplingSecondWeight_measurable h α) |
| (fun p => samplingWeight_bounds false h α (Int.fract (p.1-p.2)) p.2 hh hα) |
| have hsmall : Integrable (fun p : ℝ × ℝ => samplingTestedSecondWeight h α E G p.1 p.2 ^ 2) |
| (unitIntervalMeasure.prod unitIntervalMeasure) := |
| sampling_square_integrable _ _ (max 1 (1/(h*α))) hm |
| (fun p => ⟨(hb p.1 p.2).1,(hb p.1 p.2).2.trans |
| (samplingWeight_bounds false h α (Int.fract (p.1-p.2)) p.2 hh hα).2⟩) |
| rw [← samplingSecondWeight_square_integral h α hh hh1 hα hα1, |
| ← integral_prod _ hsmall,← integral_prod _ hbig] |
| apply integral_mono_ae hsmall hbig |
| exact Eventually.of_forall (fun p => by |
| nlinarith [(hb p.1 p.2).1,(hb p.1 p.2).2, |
| (samplingWeight_bounds false h α (Int.fract (p.1-p.2)) p.2 hh hα).1]) |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
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|
